Differentiation for Physics - Notes | Study Physics For JEE - JEE

Function: If the value of a quantity y (say) depends on the value of another quantity x, then y is the function of x i.e. y = f(x).
The quantity y is called dependent variable and the quantity x is called independent variable. For example, y = 2x² + 4x + 7 is a function of x
(i) When x = 1, y = 2(1)² + 4x1+7 = 13
(ii) When x = 2, y = 2(2)² +4x2+7 = 23
As the value of y depends on the value of x, y is the function of x.

Differential coefficient or derivative of a function
Let y = f(x) …. (1)
That is, the value of y depends upon the value of x.
Let ∆x be a small increment in x, so that ∆y is the corresponding small increment in y, then
y + ∆y = f(x+∆x) …. (2)
Subtract (1) from (2), we get ∆y = f(x+∆x) − f(x)
Divide both sides by ∆x

Where  is called average rate of change of y w.r.t. x.
Let us ∆x be as small as possible i.e. ∆x→0 (read as delta x tends to zero)
Then differential coefficient or derivative of y w.r.t. x is

Theorems of Differentiations

1. If y = C, when C is constant
2. If y = xn, where n is an integer
3. If y = Cu, where u is the function of x and C is constant
4. If y = u ± v ± ω ± ……, where u, v and ω are the function of x
   
5. If y = u v, where u and v are the function of x, then
6. If y =v/u , where u and v are the function of x, then
7. If y = uⁿ, where u is the function of x then

Differential coefficients of Trigonometric Functions

Example: Differentiate the following w.r.t. x.

(i) sin 2 x
(ii) x sin x

(i) Let y = sin 2x

(ii) Let y = x sin x 

Differential Coefficients of Logarithmic and Exponential Functions

Example: Differentiate the follow w.r.t .x.

(i) 2 (log_e x)²
(ii) log(ax + b)

(i) Let y = (log_e x)²

(ii) Let y  = log(ax + b)

Example: If S = 2t³ - 3t² + 2,find the position, velocity and acceleration of a particle at the end of 2s. S is measured in metre and t in second

S = 2t³ - 3t² + 2

When t = 2s, S = 2 × 8 – 3 × 4 + 2 = 6 m