Motion in a Plane, Chapter Notes, Class 11, Physics (IIT-JEE & AIPMT) PDF Download
Types of Vectors and Scalars - Motion in a Plane, Class 11, Physics
1. Scalar :
In physics we deal with two type of physical quantity one is scalar and other is vector. Each scalar quantity has a magnitude and a unit.
For example mass = 4kg
Magnitude of mass = 4
and unit of mass = kg
Example of scalar quantities : mass, speed, distance etc.
Scalar quantities can be added, subtracted and multiplied by simple laws of algebra.
2. Vector :
Vector are the physical quantites having magnitude as well as specified direction.
For example :
Speed = 4 m/s (is a scalar)
Velocity = 4 m/s toward north (is a vector)
If someone wants to reach some location then it is not sufficient to provide information about the distance of that location it is also essential to tell him about the proper direction from the initial location to the destination.
The magnitude of a vector () is the absolute value of a vector and is indicated by 
or A.
Example of vector quantity : Displacement, velocity, acceleration, force etc.
Knowledge of direction
3. General Points Regarding Vectors :
3.1 Representation of vector :
Geometrically, the vector is represented by a line with an arrow indicating the direction of vector as
Mathematically, vector is represented by 
.
Sometimes it is represented by bold letter A.
Thus, the arrow in abow figure represents a vector 
in xy-plane making an angle θ with x-axis.
A representation of vector will be complete if it gives us direction and magnitude.
Symbolic form : 
used to separate a vector quantity from scalar quantities (u, i, m)
Graphical form : A vector is represented by a directed straight line,having the magnitude and direction of the quantity represented by it.
e.g. if we want to represent a force of 5 N acting 45° N of E
(i) We choose direction co-ordinates.
(ii) We choose a convenient scale like 1 cm º = 1 N
(iii) We draw a line of length equal in magnitude and in the direction of vector to the chosen quantity.
(iv) We put arrow in the direction of vector.
Magnitude of vector :
3.2 Angle between two Vectors (θ)
Angle between two vectors means smaller of the two angles between the vectors when they are placed tail to tail by displacing either of the vectors parallel to itself (i.e 0 £ q £ p).
Ex.1 Three vectors 
are shown in the figure. Find angle between (i) 
and 
, (ii) 
and 
, (iii) 
and 
.
Sol. To find the angle between two vectors we connect the tails of the two vectors. We can shift 
& 
such that tails of 
and 
are connected as shown in figure.
Now we can easily observe that angle between 
and 
is 60º, 
and 
is 15º and between 
and 
is 75º.
3.3 Negative of Vector
It implies vector of same magnitude but opposite in direction.
3.4 Equality of Vectors.
Vectors having equal magnitude and same direction are called equal vectors
3.5 Collinear vectors :
Any two vectors are co-linear then one can be express in the term of other.
= 
(where l is a constant)
3.6 Co-initial vector : If two or more vector start from same point then they called co-initial vector.
e.g.
here A, B, C, D are co-initial.
3.7 Coplanar vectors :
Three (or more) vectors are called coplanar vectors if they lie in the same plane or are parallel to the same plane. Two (free) vectors are always coplanar.
Important points
A If the frame of reference is translated or rotated the vector does not change (though its components may change).
Two vectors are called equal if their magnitudes and directions are same, and they represent values of same physical quantity.
3.8 Multiplication and division of a vector by a scalar
Multiplying a vector 
with a positive number λ gives a vector 
whose magnitude become λ times but the direction is the same as that of 
. Multiplying a vector 
by a negative number λ gives a vector 
whose direction is opposite to the direction of 
and whose magnitude is -λ times 
.
The division of vector 
by a non-zero scalar m is defined as multiplication of 
by 
At here 
and 
are co-linear vector
Ex.2 A physical quantity (m = 3kg) is multiplied by a vector 
such that 
. Find the magnitude and direction of 
if
(i) 
= 3m/s2 East wards
(ii) 
= -4 m/s2 North wards
Sol. (i) 
East wards
= 9 N East wards
(ii) 
North wards
= -12 N North wards
= 12 N South wards
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Triangular and Parallelogram Laws of Addition adn Substraction of Vectors, Class 11, Physics
4. LAWS OF ADDITION AND SUBSTACTION OF VECTORS
4.1 Triangle rule of addition : Steps for additing two vector representing same physical quantity by triangle law.
(i) Keep vectors s.t. tail of one vector coincides with head of other.
(ii) Join tail of first to head of the other by a line with arrow at head of the second.
(iii) This new vector is the sum of two vectors. (also called reultant)
(i) 
(ii) 
(iii) 
Take example here.
Q. A boy moves 4 m south and then 5 m in direction 37° E of N. Find resultant displacement.
4.2 Polygon Law of addition :
This law is used for adding more than two vectors. This is extension of triangle law of addition. We keep on arranging vectors s.t. tail of next vector lies on head of former.
When we connect the tail of first vector to head of last we get resultant of all the vectors.
4.3 Parallelogram law of addition :
Steps :
(i) Keep two vectors such that there tails coincide.
(ii) Draw parallel vectors to both of them considering both of them as sides of a parallelogram.
(iii) Then the diagonal drawn from the point where tails coincide represents the sum of two vectors, with its tail at point of coincidence of the two vectors.
(i) 
(ii) 
(iii) 
Note : 
and 
thus 
[Cummutative Law]
Note : Angle between 2 vectors is the angle between their positive directions.
Suppose angle between these two vectors is θ, and 
(AD)2 = (AE)2 +(DE)2
= (AB + BE)2 + (DE)2
= (a +b cosθ)2 + (b sinθ)2
= a2 + b2 cos2θ + 2ab cosθ + b2 sin2θ
= a2 + b2 + 2ab cosθ
Thus, AD = 
or 
angle α with vector a is
tan α = 
= 
Important points :
To a vector, only a vector of same type can be added that represents the same physical quantity and the resultant is also a vector of the same type.
As R = [A2 + B2 + 2AB cosθ]1/2 so R will be maximum when, cosθ = max = 1,
i.e., θ = 0º, i.e. vectors are like or parallel and Rmax = A + B.
and angle between them θ then R = 
and angle between them π -θ then R = 
The resultant will be minimum if, cosθ = min = -1, i.e., θ = 180º, i.e. vectors are antiparallel and Rmin = A -B.
If the vectors A and B are orthogonal, i.e., θ = 90º, 
As previously mentioned that the resultant of two vectors can have any value from (A -B) to (A + B) depending on the angle between them and the magnitude of resultant decreases as q increases 0º to 180º.
Minimum number of unequal coplanar vectors whose sum can be zero is three.
The resultant of three non-coplanar vectors can never be zero, or minimum number of non coplanar vectors whose sum can be zero is four.
5. SUBTRACTION OF VECTOR :
Negative of a vector say 
is a vector of the same magnitude as vector but pointing in a direction opposite to that of 
.
Thus, 
can be written as 
or 
is really the vector addition of 
and 
.
Suppose angle between two vectors and is θ. Then angle between and 
will be 180° -θ as shown in figure.
Magnitude of 
will be thus given by
S = 
= 
or S = 
...(i)
For direction of 
we will either calculate angle α or β, where,
tanα = 
= 
...(ii)
or tanβ = 
= 
...(iii)
Ex.3 Two vectors of 10 units & 5 units make an angle of 120° with each other. Find the magnitude & angle of resultant with vector of 10 unit magnitude.
⇒
Sol. 
= 
⇒ α = 30°
[Here shows what is angle between both vectors = 120° and not 60°]
Note : 
or 
can also be found by making triangles as shown in figure. (a) and (b)
Or 
Ex.4 Two vectors of equal magnitude 2 are at an angle of 60° to each other find magnitude of their sum & difference.
Sol. 
Ex.5 Find 
and 
in the diagram shown in figure. Given A = 4 units and B = 3 units.
Sol. Addition :
R = 
= 
= 
units
tanα = 
= 
= 0.472
a = tan-1(0.472) = 25.3°
Thus, resultant of 
and 
is 
units at angle 25.3° from 
in the direction shown in figure.
Subtraction : S = 
= 
= 
and tanθ = 
= 
= 1.04
α = tan-1 (1.04) = 46.1°
Thus, 
is 
units at 46.1° from 
in the direction shown in figure.
6. Unit Vector and Zero vector
Unit vector is a vector which has a unit magnitude and points in a particular direction. Any vector 
can be written as the product of unit vector 
in that direction and magnitude of the given vector.
or 
A unit vector has no dimensions and unit. Unit vectors along the positive x-, y-and z-axes of a rectangular coordinate system are denoted by 
and respectively such that 
A vector of zero magnitude is called a zero or a null vector. Its direction is arbitrary.
Ex.6 A unit vector along East is defined as 
. A force of 105 dynes acts west wards. Represent the force in terms of 
.
Sol. 
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Resolution of Vectors - Motion in a Plane, Class 11, Physics
7. RESOLUTION OF VECTORS
If 
and 
be any two non-zero vectors in a plane with different directions and 
be another vector in the same plane. 
can be expressed as a sum of two vectors-one obtained by multiplying by a real number and the other obtained by multiplying by another real number.
(where l and m are real numbers)
We say that 
has been resolved into two component vectors namely
(where l and m are real number)
We say that 
has been resolved into two component vectors namely
and 
along 
and 
respectively. Hence one can resolve a given vector into two component vectors along a set of two vectors - all the three lie in the same plane.
7.1 Resolution along rectangular component :
It is convenient to resolve a general vector along axes of a rectangular coordinate system using vectors of unit magnitude, which we call as unit vectors. 
are unit along x, y and z-axis as shown in figure below :
7.2 Resolution in two Dimension
Consider a vector 
that lies in xy plane as shown in figure,
⇒ 
The quantities Ax and Ay are called x-and y-components of the vector 
.
Ax is itself not a vector but 
is a vector and so it 
.
Ax = A cosθ and Ay = A sinθ
It's clear from above equation that a component of a vector can be positive, negative or zero depending on the value of q. A vector 
can be specified in a plane by two ways :
(a) its magnitude A and the direction q it makes with the x-axis; or
(b) its components Ax and Ay A = 
, θ = 
Note : If A = Ax ⇒ Ay = 0 and if A = Ay ⇒ Ax = 0 i.e., components of a vector perpendicular to itself is always zero. The rectangular components of each vector and those of the sum 
are shown in figure.
We saw that
is equivalent to both
Cx = Ax + Bx
and Cy = Ay + By
Refer figure (b)
Vector 
has been resolved in two axes x and y not perpendicular to each other. Applying sine law in the triangle shown, we have
or Rx = 
and Ry = 
If α+β = 90°, Rx = R sinβ and Ry = R sin
Ex.7 Resolve the vector 
along an perpendicular to the line which make angle 60° with x-axis.
Sol. 
Ex.8 Resolve a weight of 10 N in two directions which are parallel and perpendicular to a slope inclined at 30° to the horizontal
Sol. Component perpendicular to the plane
= 
= 
N Ans.
and component parallel to the plane
W|| =W sin 30° = (10) 
= 5 N
Ex.9 Resolve horizontally and vertically a force F = 8 N which makes an angle of 45° with the horizontal.
Sol. Horizontal component of is 
FH = F cos 45° = (8) 
= 
and vertical component of is
Fv = F sin 45° = 
= 
Ans.
8. PROCEDURE TO SOLVE THE VECTOR EQUATION
...(1)
(a) There are 6 variables in this equation which are following :
(1) Magnitude of 
and its direction
(2) Magnitude of 
and its direction
(3) Magnitude of 
and its direction.
(b) We can solve this equation if we know the value of 4 variables [Note : two of them must be directions]
(c) If we know the two direction of any two vectors then we will put them on the same side and other on the different side.
For example
If we know the directions of 
and and 
direction is unknown then we make equation as follows:-
(d) Then we make vector diagram according to the equation and resolve the vectors to know the unknown values.
Ex.10 Find the net displacement of a particle from its starting point if it undergoes two sucessive displacement given by 
, 37° North of West, 
, 53° North of East
Sol. 
Ex.11 Find magnitude of 
and direction of 
. If 
makes angle 37° and 
makes 53° with x axis and 
has magnitude equal to 10 and 
has 5. (given 
)
Sol.
Ex.12 Find the magnitude of F1 and F2. If F1, F2 make angle 30° and 45° with F3 and magnitude of F3 is 10 N. (given 
= 
)
Sol.
Means component of 
perpendicular to resultant is equal in magnitude to the component of 
perpendicular to resultant.
Ex.13 If two vectors 
and 
make angle 30° and 45° with their resultant and 
has magnitude equal to 10, then find magnitude of 
.
Sol. B sin 60° = A sin 30°
⇒ 10 sin 60° = A sin 30°
⇒ A = 
Ex.14 If 
and 
have angle between them equals to 60° and their resultant make, angle 45° with 
and 
have magnitude equal to 10. Then Find magnitude of 
.
Sol. here a = 45° and b = 60° -45° = 15°
so A sinα = B sinβ
10 sin 45° = B sin 45°
So B = 
= 
10. ADDITION AND SUBTRACTION IN COMPONENT FORM :
Suppose there are two vectors in component form. Then the addition and subtraction between these two are
Also if we are having a third vector present in component form and this vector is added or subtracted from the addition or subtraction of above two vectors then
Note : Modulus of vector A is given by
Ex.15 Obtain the magnitude of 
if
and 
Sol. 
Magnitude of 
= 
Ans.
Ex.16 Find 
and 
if 
make angle 37° with positive x-axis and 
make angle 53° with negative x-axis as shown and magnitude of 
is 5 and of B is 10.
Sol. 
for 
+ 
= 
so the magnitude of resultant will be = 
= 
and have angle θ = 
from negative x - axis towards up
for 
So the magnitude of resultant will be
= 
and have angle 
from positive x-axis towards down.
11. MULTIPLICATION OF VECTORS (The Scalar and vector products) :
11.1 Scalar Product
The scalar product or dot product of any two vector 
and 
, denoted as 
. 
(read 
dot 
) is defined as the product of their magnitude with cosine of angle between them.
Thus,
(here θ is the angle between the vectos)
Properties :
- It is always a scalar which is positive if angle between the vectors is acute (i.e.< 90°) and negative if angle between them is obtuse (i.e., 90° < q £ 180°)
- It is commutative i.e.
- It is distributive, i.e.
- As by definition . = AB cosθ . The angle between the vectors θ =
.
= AB cosθ . The angle between the vectors θ = 
Geometrically, B cosθ is the projection of 
onto 
and vice versa
Component of 
along 
= B cosθ = 
= 
(Projection of 
on 
)
Component of 
along 
= A cosθ = 
= 
(Projection of 
on 
)
- Scalar product of two vectors will be maximum when cosθ = max = 1, i.e., θ = 0°,
i.e., vectors are parallel ⇒ 
- If the scalar product of two non-zero vectors vanishes then the vectors are perpendicular.
- The scalar product of a vector by itself is termed as self dot product and is given by
= AA cosθ = A2 ⇒ 
- In case of unit vector ,
, 
Ex.17 If the vectors 
and 
are perpendicular to each other. Find the value of a?
Sol. If vectors 
and 
are perpendicular
⇒ 
⇒ 
⇒ a2 -2a -3 = 0 ⇒ a2 -3a a -3 = 0
⇒ a(a -3) +1 (a -3 ) ⇒ a = -1, 3
Ex.18 Find the component of 
along 
?
Sol. Component of 
along 
is given by 
hence required component
= 
Ex.19 Find angle between 
and 
?
Sol. We have cosθ = 
cosθ = 
= 
θ = cos-1 
Ex.20 (i) For what value of m the vector 
is perpendicular to 
(ii) Find the component of vector 
along the direction of 
?
Sol. (i) m = -10 (ii) 
Important Note :
Components of b along and perpendicular to a.
Let 
. 
represent two (non-zero) given vectors a, b respectively. Draw BM perpendicular to 
From ΔOMB, 
= 
Ex.21 The velocity of a particle is given by 
. Find the vector component of its velocity parallel to the line 
.
Sol. Component of 
along 
11.2 Vector product
The vector product or cross product of any two vectors and 
, denoted as
(read 
cross 
) is defined as :
Here θ is the angle between the vectors and the direction 
is given by the right - hand - thumb rule.
Right - Hand - Thumb Rule :
To find the direction of 
, draw the two vectors 
and 
with both the tails coinciding. Now place your stretched right palm perpendicular to the plane of 
and 
in such a way that the fingers are along the vector 
and when the fingers are closed they go towards 
. The direction of the thumb gives the direction of 
.
Properties :
- Vector product of two vectors is always a vector perpendicular to the plane containing the two vectors i.e. orthogonal to both the vectors and , though the vectors and may or may not be orthogonal.
- Vector product of two vectors is not commutative i.e. But
- The vector product is distributive when the order of the vectors is strictly maintained i.e.
- The magnitude of vector product of two vectors will be maximum when sinθ = max = 1. i.e. θ = 90°
- The magnitude of vector product of two non-zero vectors will be minimum when |sinθ| = minimum = 0, i.e., θ = 0° or 180° and i.e., if the vector product of two non-zero vectors vanishes, the vectors are collinear.
- The self cross product i.e. product of a vector by itself vanishes i.e. is a null vector.
- In case of unit vector , ⇒
- In case of orthogonal unit vectors and in accordance with right-hand-thumb-rule,
may or may not be orthogonal.
But 
i.e., if the vector product of two non-zero vectors vanishes, the vectors are collinear.
, 
⇒ 
and 
in accordance with right-hand-thumb-rule, 
Ex.22 
is East wards and 
is downwards. Find the direction of 
× 
?
Sol. Applying right hand thumb rule we find that 
is along North.
Ex.23 If 
, find angle between 
and 
Sol. 
AB cosθ = AB sinθ tanθ = 1 ⇒ θ = 45°
Ex.24 
⇒ 
here 
is perpendicular to both 
and 
Ex.25 Find 
if 
and 
Sol. 
= 
= 
Ex.26 (i) 
is North-East and 
is down wards, find the direction of 
(ii) Find 
× 
if 
and 
Ans. (i) North - West. (ii) 
12. POSITION VECTOR :
Positin vector for a point is vector for which tail is origin & head is the given point itself.
Position vector of a point defines the position of the point w.r.t. the origin.
CALCULUS
14. Constants : They are fixed real number which value does not change
Ex. 3, e, a, -1, etc.
15. Variable :
Somthing that is likely to vary, somthing that is subject to variation.
or
A quantity that can assume any of a set of value.
Types of variables.
(i) Independent variables : Indepedent variables is typically the variable being manipulated or change
(ii) dependent variables : The dependent variables is the object result of the independent variable being manipulated.
Ex. y = x2
here y is dependent variable and x is independent variable
16. FUNCTION :
Function is a rule of relationship between two variables in which one is assumed to be dependent and the other independent variable.
The temperatures at which water boils depends on the elevation above sea level (the boiling point drops as you ascend). Here elevation above sea level is the independent & temperature is the dependent variable.
The interest paid on a cash investment depends on the length of time the investment is held. Here time is the independent and interest is the dependent variable.
In each case, the value of one variable quantity (dependent variable), which we might call y, depends on the value of another variable quantity (independent variable), which we might call x. Since the value of y is completely determined by the value of x, we say that y is a function of x and represent it mathematically as y = f(x).
all possible values of independent variables (x) are called domain of function.
all possible values of dependent variable (y) are called Range of fucntion.
Think of function f as a kind machine that produces an output value f(x) in its range whenever we feed it an input value x from its domain (figure).
When we study circles, we usualy call the area A and the radius r. Since area depends on radius, we say that A is a function of r, A = f(r). The eauation A = πr2 is a rule that tells how to calculate a unique (single) output value of A for each possible input value of the radius r.
A = f(x) = πr2. (Here the rule of relationship which describes the function may be described as square & multiply by π)
if r = 1 A = π
if r = 2 A = 4π
if r = 3 A = 9π
The set of all possible input values for the radius is called the domain of the function. The set of all output values of the area is the range of the function.
We usually denote functions in one of the two ways :
1. By giving a formula such as y = x2 that uses a dependent variable y to denote the value of the fucntion.
2. By giving a formula such as f(x) =x2 that defines a functions symbols f to name the function.
Strictly speaking, we should call the function f and not f(x).
y = sinx. Here the function is y since, x is the independent variable.
Ex.27 The volume V of ball (solid sphere) of radius r is given by the function V(r) = 
The volume of a ball of radius 3m is ?
Sol. V(3) = 
= 36 pm3.
Ex.28 Suppose that the function F is defined for all real numbers r by the formula.
F(r) = 2 (r -1) +3.
Evaluate F at the input values 0, 2 x 2, and F(2).
Sol. In each case we substitute the given input value for r into the formula for F:
F(0) = 2(0 -1) + 3 = -2 + 3 = 1
F(2) = 2(2 -1) + 3 = 2 + 3 =5
F(x + 2) = 2 (x + 2 -1) + 3 = 2x + 5
F(F(2)) = F(5) = 2(5 -1) 3 = 11
17. Differentiation
Finite difference :
The finite difference between two values of a physical is represented by Δ notation.
For example :
Difference in two values of y is written as Δy as given in the table below.
Infinitely small difference :
The infinitely small difference means very-very small difference. And this difference is represented by 'd' notation insted of 'D'.
For example infinitely small difference in the values of y is written as 'dy'
if y2 = 100 and y1 = 99.9999999999999.....
then dy = 0.00000000000000..........00001
Definition of differentiation
Another name of differentiation is derivative. Suppose y is a function of x or y = f(x)
Differentiation of y with respect to x is denoted by sumbols f' (x)
where f'(x) = 
; dx is very small change in x and dy is corresponding very small change in y.
Notation : There are many ways to denote the derivative of function y = f(x), the most common notations are these :
Average rates of change :
Given an arbitrary function y = f(x) we calculate the average rate of change of y with respect to x over the interval (x, x +Δx) by dividing the change in value of y, i.e., Dy = f(x+Δx) -f(x), by length of interval Δx over which the change occurred.
The average rate of change of y with respect to x over the interval [x, x+Δx]
Geometrically
= tanθ = Slope of the line PQ
In triangle QPR tanθ = 
therefore we can say that average rate of change of y with respect to x is equal to slope of the line joining P & Q.
The derivative of a fucntion
We know that Average rate of change of y w.r.t x is -
If the limit of this ratio exists as Δx → 0, then it is called the derivative of given function f(x) and is denoted as
18. GEOMETRICAL MEANING OF DIFFERENTIATION :
The geometrical meaning of differentiation is very much useful in the analysis of graphs in physics. To understand the geometrical meaning of derivatives we should have knowledge of secant and tangent to a curve.
Secant and Tangent to a Curve
Secant : - A secant to a curve is a straight line, which intersects the curve at any two points.
Tangent :
A tangent is straight line, which touches the curve a particular point. Tangent is limiting case of secant which intersects the curve at two overlapping point.
In the figure - 1 shown, if value of Δx is gradually reduced then the point Q will move nearer to the point P. If the process is continuously repeated (Figure-2) value of Δx will be infinitely small and secant PQ to the given curve will become a tangent at point P.
Therefore
we can say that differentiation of y with respect to x, i.e. 
is equal to slope of the tangent at point P (x,y)
or tanθ = 
(From fig-1 the average rate change of y from x to x+Δx is identical with the slope of secant PQ)
Rule No. 1 Derivative Of A Constant
The first rule of differentiation is that the derivative of every constant function is zero.
If c is constant, then 
Ex.30 
, 
, 
Rule No.2 Power Rule
If n is a real number, then 
To apply the power Rule, we subtract 1 from the original exponent (n) and multiply the result by n.
Ex.31
Rule No.3 The Constant Multiple Rule
If u is a differentiable function of x, and c is a constant, then 
In particular, if n is a positive integer, then 
Ex.34 The derivative formula
says that if we rescale the graph of y = x2 by multiplying each y-coordinate by 3, then we multiply the slope at each point by 3.
Ex.35 A useful special case
The derivative of the negative of a differentiable function is the negative of the function's derivative. Rule 3 with c = -1 gives.
Rule No.4 The Sum Rule
The derivative of the sum of two differentiable functions is the sum of their derivatives.
If u and v are differentiable functions of x, then their sum u+v is differentiable at every point where u and v are both differentiable functions in their derivatives.
The sum Rule also extends to sums of more than two functions, as long as there are only finite functions in the sum. If u1, u2, ........ un are differentiable at x, then so if u1+u2 ....... +un, then
Notice that we can differentiate any polynomial term by term, the way we differentiated the polynomials in above example.
Rule No. 5 The Product Rule
If u and v are differentiable at x, then if their product uv is considered, then 
.
The derivative of the product uv is u times the derivative of v plus v times the derivative of u. In prime notation
(uv)' = uv' + vu'.
While the derivative of the sum of two functions is the sum of their derivatives, the derivative of the product of two functions is not the product of their derivatives. For instance,
while 
, which is wrong
Ex.37 Find the derivatives of y = (x2+1) (x3+3)
Sol. Using the product Rule with u = x2+1 and v = x3+3, we find
= (x2+1) (3x2) + (x3+3) (2x)
= 3x4 + 3x2 + 2x4 + 6x = 5x4 + 3x2 + 6x
Example can be done as well (perhaps better) by multiplying out the original expression for y and differentiating the resulting polynomial. We now check :
y = (x2 + 1) (x3 + 3) = x5 + x3 + 3x2 + 3
= 5x4 + 3x2 + 6x
This is in agreement with our first calculation.
There are times, however, when the product Rule must be used. In the following examples. We have only numerical values to work with.
Ex.38 Let y = uv be the product of the functions u and v. Find y'(2) if u(2) = 3, u'(2) = -4, v(2) = 1, and v'(2) = 2.
Sol.
Rule No.6 The Quotient Rule
If u and v are differentiable at x, and v(x) ¹ 0, then the quotient u/v is differentiable at x,
and 
Just as the derivative of the product of two differentiable functions is not the product of their derivatives, the derivative of the quotient of two functions is not the quotient of their derivatives.
Ex.39 Find the derivative of 
Sol. We apply the Quotient Rule with u = t2 -1 and v = t2 1
Rule No. 7 Derivative Of Sine Function
Rules No.8 Derivative Of Cosine Function
Rule No. 9 Derivatives Of Other Trigonometric Functions
Because sin x and cos x are differentiable functions of x, the related functions
; 
; 
are differentiable at every value of x at which they are defined. There derivatives, Calculated from the Quotient Rule, are given by the following formulas.
; 
; 
Ex.42 Find dy / dx if y = tan x.
Sol. 
Rule No. 10 Derivative Of Logrithm And Exponential Functions
, 
Ex.44 y = ex . loge (x)
⇒ 
Rule No. 11 Chain Rule Or `Outside Inside' Rule
It sometime helps to think about the Chain Rule the following way. If y = f(g(x)),
= f'[g(x)] . g'(x)
In words : To find dy/dx, differentiate the "outside" function f and leave the "inside" g(x) alone; then multiply by the derivative of the inside.
We now know how to differntiate sin x and x2 -4, but how do we differentiate a composite like sin(x2 -4)?
The answer is, with the Chain Rule, which says that the derivative of the composite of two differentiable functions is the product of their derivatives evaluated at appropriate points. The Chain Rule is probably the most widely used differentiation rule in mathematics. This section describes the rule and how to use it. We begin with examples.
Ex.45 The function y = 6x -10 = 2(3x -5) is the composite of the functions y = 2u and u = 3x -5. How are the derivatives of these three functions related ?
Sol. We have 
, 
, 
Since 6 = 2 × 3 
Is it an accident that 
?
If we think of the derivative as a rate of change, our intution allows us to see that this relationship is reasonable. For y = f(u) and u = g(x), if y changes twice as fast as u and u changes three times as fast as x, then we expect y to change six times as fast as x.
Ex.46 Let us try this again on another function.
y = 9x4 +6x2 +1 = (3x2 +1)2
is the composite y = u2 and u = 3x2 + 1. Calculating derivatives. We see that
= 2 (3x2 + 1). 6x = 36x3 + 12 x
and 
= 36 x3 + 12 x
Once again, 
The derivative of the composite function f(g(x)) at x is the derivative of f at g(x) times the derivative of g at x.
Ex.47 Find the derivation of 
Sol. Here y = f(g(x)), where f(u) = 
and u = g(x) = x2 + 1. Since the derivatives of f and g are
f' (u) = 
and g'(x) = 2x,
the Chain Rule gives
= f' (g(x)).g'(x) = 
.g'(x) = 
. (2x) = 
Ex.48
Rull No. 12 Power Chain Rule
* If 
Ex.50 
= 
= -1 (3x -2)-2 
= -1 (3x -2)-2 (3) = -
In part (d) we could also have found the derivation with the Quotient Rule.
Ex.51 (a) 
Sol. Here u = Ax B, 
(b) 
(c) 
log(Ax B) = 
.A
(d) 
tan (Ax + B) = sec2 (Ax + B).A (e) 
Note : These results are important
19. DOUBLE DIFFERENTIATION
If f is differentiable function, then its derivative f' is also a function, so f' may have a derivative of its own, denoted by 
. This new function f'' is called the second derivative of because it is the derivative of the derivative of f. Using Leibniz notation, we write the second derivative of y = f(x) as
Another notation is f''(x) = D2 f(x).
20. Application of derivative Differentiation as a rate of change
is rate of change of 'y' with respect to 'x' :
For examples :
(i) v = 
this means velocity 'v' is rate of change of displacement 'x' with respect to time 't'
(ii) a = 
this means acceleration 'a' is rate of change of velocity 'v' with respect to time 't'.
(iii) 
this means force 'F' is rate of change of monentum 'p' with respect to time 't'.
(iv) 
= 
this means torque 't' is rate of change of angular momentum 'L' with respect to time 't'
(v) Power = 
this means power 'P' is rate of change of work 'W' with respect to time 't'
Ex.53 The area A of a circle is related to its diameter by the equation 
.
How fast is the area changing with respect to the diameter when the diameter is 10 m ?
Sol. The (instantaneous) rate of change of the area with respect to the diameter is
When D =10m, the area is changing at rate (π/2) = 5π m2/m. This mean that a small change ΔD m in the diameter would result in a changed of about 5p ΔD m2 in the area of the circle.
Physical Example :
Ex.54 Boyle's Law state that when a sample of gas is compressed at a constant temperature, the product of the pressure and the volume remains constant : PV = C. Find the rate of change of volume with respect to pressure.
Sol. 
Ex.55 (a) Find the average rate of change of the area of a circle with respect to its radius r as r changed from
(i) 2 to 3 (ii) 2 to 2.5 (iii) 2 to 2.1
(b) Find the instantaneous rate of change when r = 2.
(c) Show that thre rate of change of the area of a circle with respect to its radius (at any r) is equal to the circumference of the circle. Try to explain geometrically when this is true by drawing a circle whose radius is increased by an amount Δr. How can you approximate the resulting change in area ΔA if Δr is small ?
Sol. (a) (i) 5π (ii) 4.5 π (iii) 4.1 π
(b) 4π
(c) ΔA ≈ 2 πrΔr
21. MAXIMA & MINIMA
Suppose a quantity y depends on another quantity x in a manner shown in figure. It becomes maximum at x1 and minimum at x2. At these points the tangent to the curve is parallel to the x-axis and hence its slope is tanθ = 0. Thus, at a maxima or a minima slope
⇒ 
Maxima
Just before the maximum the slope is positive, at the maximum it is zero and just after the maximum it is negative. Thus, 
decrease at a maximum and hence the rate of change of 
is negative at a maximum i.e., 
at maximum. The quantity 
is the rate of change of the slope. It is written
as 
. Conditions for maxima are : (a) 
(b) 
Minima
Similarly, at a minimum the slope changes from negative to positive, Hence with the increases of x. The slope is increasing that means the rate of change of slope with respect to x is positive.
Hence 
Conditions for minima are :
(a) 
(b) 
Quite often it is known from the physical situation whether the quantity is a maximum or a minimum. The test on 
may then be omitted.
Ex.56 Find maximum or minimum values of the functions :
(A) y = 25x2 + 5 -10x (B) y = 9 -(x -3)2
Sol. (A) For maximum and minimum value, we can put 
or 
x = 
Further, 
or 
has positive value at x = 
. Therefore, y has minimum value at x = 
. Therefore, y has minimum value at x = 
. Substituting x =
in given equation, we get
ymin = 
(B) y = 9 -(x -3)2 = 9 -x2 +-9 6x
or y = 6x -x2
For minimum or maximum value of y we will substitute 
or 6 -2x = 0
x = 3
To check whether value of y is maximum or minimum at x = 3 we will have to check whether 
is positive or negative.
or 
is negative at x = 3. Hence, value of y is maximum. This maximum value of y is,
ymax = 9 -(3 -3)2 = 9
22. INTEGRATION
Definitions :
A function F(x) is a antiderivative of a function f(x) if
F'(x) = f(x)
for all x in the domain of f. The set of all antiderivatives of f is the indefinite integral of f with respect to x, denoted by
The symbol 
is an integral sign. The function f is the integrand of the integral and x is the variable of integration.
For example f(x) = x3 then f'(x) = 3x2
So the integral of 3x2 is x3
Similarly if f(x) = x3 + 4
there for general integral of 3x2 is x3 + c where c is a constant
One antiderivative F of a function f, the other antiderivatives of f differ from F by a constant. We indicate this in integral notation in the following way :
.....(i)
The constant C is the constant of integration or arbitrary constant, Equation (1) is read, "The indefinite integral of f with respect to x is F(x) + C." When we find F(x) + C, we say that we have integrated f and evaluated the integral.
Ex.57 Evaluate 
Sol. 
The formula x2 + C generatres all the antiderivatives of the function 2x. The function x2 + 1, x2 -π, and
x2+ 
are all antiderivatives of the function 2x, as you can check by differentiation.
Many of the indefinite integrals needed in scientific work are found by reversing derivative formulas.
Integral Formulas
Indefinite Integral Reversed derivated formula
1. 
,n ¹ -1, n rational 
= xn
(special case) 
2. 
3. 
4. 
5. 
6. 
= sec x tan x
7. 
= -cosec x +C 
Ex.58 Examples based on above formulas :
(a) 
(b) 
Formula 1 with n = 5
(c) 
Formula 1 with n = 
(d) 
Formula 2 with k = 2
(e) 
= 
= 
Formula 3 with k = 
Ex.59 Right : 
= x sin x + cos x C
Reason : The derivative of the right-hand side is the integrand :
Check : 
= x cos x + sin x -sin x + 0 = x cos x.
Wrong : 
= x sin x +C
Reason : The derivative of the right-hand side is not the integrand :
Check : 
= x cos x + sin x + 0 
x cos x
Rule No. 1 Constant Multiple Rule
- A function is an antiderivative of a constant multiple k of a function f if and only if it is k times an antiderivative of f.
Ex.60 
= 
= 
Rule No.2 Sum And Difference Rule
- A function is an antiderivative of a sum or difference f ± g if and only if it is the sum or difference of an antiderivative of f an antiderivative of g.
Ex.61 Term-by-term integration
Evaluate : 
Sol. If we recognize that (x3/3) -x2 5x is an antiderivative of x2 -2x +5, we can evaluate the integral as
If we do not recognize the antiderivative right away, we can generate it term by term with the sum and difference Rule :
This formula is more complicated than it needs to be. If we combine C1, C2 and C3 into a single constant
C = C1 + C2 + C3, the formula simplifies to
and still gives all the antiderivatives there are. For this reason we recommend that you go right to the final form even if you elect to integrate term by term. Write
Find the simplest antiderivative you can for each part add the constant at the end.
Ex.62 We can sometimes use trigonometric identities to transform integrals we do not know how to evaluate into integrals. The inetgral formulas for sin2 x and cos2 x arise frequently in applications.
(a) 
= 
= 
(b) 
= 
As in part (a), but with a sign change
23. Some Indefinite integrals (An arbitrary constant should be added to each of these integrals.
(a) 
(provided n ¹ --1) C (b) 
(c) 
(d) 
(e) 
(f)
Ex.63 (a) 
(b)
(c) 
(d)
(e) 
(f) 
(g) 
(h) 
24. DEFINITE INTEGRATION OR INTEGRTION WITH LIMITS
Ex.64 
= 3 [4 -(-1)] = (3) (5) = 15
= 
+ cos (0) = -0 + 1 = 1
Ex.65 (1) 
(2) 
(3) 
25. APPLICATION OF DEFINITE INTERGRAL
Calculation Of Area Of A Curve.
From graph shown in figure if we divide whole area in infinitely small strips of dx width.
We take a strip at x position of dx width.
Small area of this strip dA = f(x) dx
So, the total area between the curve and x-axis = sum of area of all strips = 
Let f(x) > 0 be continuous on [a,b]. The area of the region between the graph of f and the x-axis is
Ex.66 Using an area to evaluate a definite integral
Evaluate 
0 < a < b.
Sol. We sketch the region under the curve y = x, a £ x £ b (figure) and see that it is a trapezoid with height (b -a) and bases a and b.
The value of the integral is the area of this trapezoid :
Thus =
and so on.
Notice that x2/2 is an antiderivative of x, further evidence of a connection between antiderivatives and summation.
(i) To find impulse
so imples = 
Ex.67 If F = kt then find impulse at t = 3 sec.
so impulse will be area under f - t curve
= 
⇒ 
2. To calculate work done by force :
So area under f - x curve will give the value of work done.
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