NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Class 11

NCERT QUESTION

 

( Limits And Derivatives )

 

Question 1:Evaluate the Given limit:

 

ANSWER :      

 

 

Question 2:Evaluate the Given limit:

 

ANSWER :      

 

 

Question 3:     Evaluate the Given limit:

 

ANSWER :      

 

 

Question 4:     Evaluate the Given limit:

 

ANSWER :

 

 

Question 5:     Evaluate the Given limit:

 

ANSWER :      

 

Question 6:     Evaluate the Given limit:

 

ANSWER :      

Put x  1 = y so that y → 1 as x → 0.

 

 

Question 7:     Evaluate the Given limit:

 

ANSWER :      

 

At x = 2, the value of the given rational function takes the form .

 

 

Question 8:     Evaluate the Given limit:

 

ANSWER :      

At x = 2, the value of the given rational function takes the form .

 

 

Question 9:     Evaluate the Given limit:

 

ANSWER :      

 

 

Question 10:   Evaluate the Given limit: 

 

ANSWER :      

 

At z = 1, the value of the given function takes the form .

Put  so that z →1 as x → 1.

 

 

Question 11:   Evaluate the Given limit:

 

ANSWER :      

 

Question 12:   Evaluate the Given limit:

 

ANSWER :      

 

At x = –2, the value of the given function takes the form .

 

 

Question 13:   Evaluate the Given limit:

 

ANSWER :      

 

At x = 0, the value of the given function takes the form .

 

 

Question 14:   Evaluate the Given limit:

 

ANSWER :      

 

At x = 0, the value of the given function takes the form .

 

 

Question 15:   Evaluate the Given limit:

 

ANSWER :      

 

It is seen that x → π ⇒ (π – x) → 0

 

 

Question 16:   Evaluate the given limit: 

 

 ANSWER :     

 

Question 17:   Evaluate the Given limit:

 

 ANSWER :     

At x = 0, the value of the given function takes the form .

Now,

 

Question 18:   Evaluate the Given limit:

 ANSWER :     

 

At x = 0, the value of the given function takes the form .

Now,

 

 

Question 19:   Evaluate the Given limit:

 

 ANSWER :     

 

 

Question 20:   Evaluate the Given limit:

 

 ANSWER :     

At x = 0, the value of the given function takes the form .

Now,

 

 

Question 21:   Evaluate the Given limit:

 

 ANSWER :      At x = 0, the value of the given function takes the form .

Now,

 

 

Question 22:  

 

ANSWER :      

 

At, the value of the given function takes the form .

Now, put  so that .

 

 

Question 23:   Find f(x) and f(x), where f(x) =

 

ANSWER :      

The given function is

f(x) =

 

 

 

 

 

 

Question 24:   Find f(x), where f(x) =

 

ANSWER :      

The given function is

 

 

Question 25:   Evaluate f(x), where f(x) = 

 

ANSWER :      

The given function is

f(x) = 

 

 

 

Question 26:   Find f(x), where f(x) =

 

ANSWER :      

The given function is

 

 

 

Question 27:   Find f(x), where f(x) =

 

ANSWER :      

The given function is f(x) =.

 

 

Question 28: Suppose f(x) =  and if f(x) = f(1) what are possible values of a and b?

 

ANSWER :      

The given function is

 

Thus, the respective possible values of a and b are 0 and 4.

 

Question 29:   Let be fixed real numbers and define a function

 

What is f(x)? For some  compute f(x).

 

ANSWER :      

The given function is

 

 

Question 30:   If f(x) =

For what value (s) of a does f(x) exists?

 

ANSWER :      

The given function is

 

When a < 0,

 

 

 

When a > 0

 

Thus,   exists for all a ≠ 0.

 

Question 31:   If the function f(x) satisfies , evaluate .

 

ANSWER :      

 

 

Question 32:

If For what integers m and n does  and  exist?

 

ANSWER :       The given function is

 

 

Thus,  exists if m = n.

 

Thus,  exists for any integral value of m and n.

 

Question 33:   Find the derivative of x² – 2 at x = 10.

 

ANSWER :       Let f(x) = x² – 2. Accordingly,

 

Thus, the derivative of x² – 2 at x = 10 is 20.

 

Question 34:   Find the derivative of 99x at x = 100.

ANSWER :      

Let f(x) = 99x. Accordingly,

 

Thus, the derivative of 99x at x = 100 is 99.

 

Question 35:   Find the derivative of x at x = 1.

ANSWER :      

Let f(x) = x. Accordingly,

 

Thus, the derivative of x at x = 1 is 1.

 

Question 36 :  Find the derivative of the following functions from first principle.

(i) x³ – 27                               (ii) (x – 1) (x – 2)

(ii)                                     (iv)

 

ANSWER :      

 (i) Let f(x) = x³ – 27. Accordingly, from the first principle,

 

(ii) Let f(x) = (x – 1) (x – 2). Accordingly, from the first principle,

 

(iii) Let . Accordingly, from the first principle,

 

(iv) Let . Accordingly, from the first principle,

 

 

 

Question 37:   For the function

 

Prove that 

 

ANSWER :      

The given function is

 

Thus,

 

Question 38:   Find the derivative of for some fixed real number a.

 

ANSWER :      

Let 

 

 

Question 39:   For some constants a and b, find the derivative of

(i) (x – a) (x – b)                   (ii) (ax² +  b)²                          (iii)

 

ANSWER :      

 (i) Let f (x) = (x – a) (x – b)

 

(ii) Let 

 

(iii)

 

By quotient rule,

 

 

Question 40:   Find the derivative of for some constant a.

 

ANSWER :      

 

By quotient rule,

 

 

Question 41:   Find the derivative of

(i)                                                                     (ii) (5x³ + 3x – 1) (x – 1)

(iii) x^–3 (5+ 3x)                                                         (iv) x⁵ (3 – 6x^–9)

(v) x^–4 (3 – 4x^–5)                                                         (vi)

 

ANSWER :      

 (i) Let

 

(ii) Let f (x) = (5x³ + 3x – 1) (x – 1)

By Leibnitz product rule,

 

(iii) Let f (x) = x^{– 3} (5 +3x)

By Leibnitz product rule,

 

(iv) Let f (x) = x⁵ (3 – 6x^–9)

By Leibnitz product rule,

 

(v) Let f (x) = x^–4 (3 – 4x^–5)

By Leibnitz product rule,

 

(vi) Let f (x) = 

 

By quotient rule,

 

 

Question 42:   Find the derivative of cos x from first principle.

ANSWER :      

Let f (x) = cos x. Accordingly, from the first principle,

 

 

 

Question 43:   Find the derivative of the following functions:

(i) sin x cos x              (ii) sec x                                (iii) 5 sec x + 4 cos x                        (iv) cosec x       (v) 3cot x + 5cosec x      (vi) 5sin x – 6cos x  7        (vii) 2tan x – 7sec x

 

ANSWER :      

 (i) Let f (x) = sin x cos x. Accordingly, from the first principle,

 

(ii) Let f (x) = sec x. Accordingly, from the first principle,

 

(iii) Let f (x) = 5 sec x+  4 cos x. Accordingly, from the first principle,

 

 

(iv) Let f (x) = cosec x. Accordingly, from the first principle,

 

(v) Let f (x) = 3cot x + 5cosec x. Accordingly, from the first principle,

 

 

From (1), (2), and (3), we obtain

 

(vi) Let f (x) = 5sin x – 6cos x  7. Accordingly, from the first principle,

 

(vii) Let f (x) = 2 tan x – 7 sec x. Accordingly, from the first principle,

 

 

Question 44:

Find the derivative of the following functions from first principle:

(i) –x (ii) (–x)^–1 (iii) sin (x + 1)

(iv)

 

ANSWER :      

 (i) Let f(x) = –x. Accordingly,

By first principle,

 

(ii) Let . Accordingly,

By first principle,

 

 

(iii) Let f(x) = sin (x + 1). Accordingly,

By first principle,

 

 

(iv) Let . Accordingly,

By first principle,

 

 

 

Question 45:   Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (x +  a)

 

ANSWER :      

Let f(x) = x  + a. Accordingly,

By first principle,

 

 

Question 46:   Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): 

 

ANSWER :      

By Leibnitz product rule,

 

 

Question 47:   Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (ax  + b) (cx +  d)²

 

ANSWER :      

Let

By Leibnitz product rule,

 

 

Question 48:   Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): 

 

ANSWER :       Let

By quotient rule,

 

 

Question 49:   Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

 

ANSWER :      

 

By quotient rule,

 

 

Question 50:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): 

 

ANSWER :      

Let

By quotient rule,

 

 

Question 51:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): 

 

ANSWER :      

 

By quotient rule,

 

 

Question 52:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): 

 

ANSWER :      

 

By quotient rule,

 

 

Question 53:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): 

 

ANSWER :      

 

 

Question 54:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): 

 

ANSWER :      

 

 

Question 55:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (ax +  b)ⁿ

 

ANSWER :      

 

By first principle,

 

 

Question 56:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (ax +  b)ⁿ (cx +  d)^m

 

ANSWER :      

Let

By Leibnitz product rule,

 

 

Therefore, from (1), (2), and (3), we obtain

 

 

Question 57:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): sin (x +  a)

 

ANSWER :      

Let

 

By first principle,

 

 

Question 58:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): cosec x cot x

 

ANSWER :      

Let

By Leibnitz product rule,

 

 

By first principle,

 

Now, let f₂(x) = cosec x. Accordingly,

By first principle,

 

 

From (1), (2), and (3), we obtain

 

 

Question 59:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): 

 

ANSWER :      

Let

By quotient rule,

 

 

Question 60:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): 

 

ANSWER :      

Let

By quotient rule,

 

 

Question 61:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): 

 

ANSWER :      

Let

 

By quotient rule,

 

 

Question 62:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): sinⁿ x

 

ANSWER :      

Let y = sinⁿ x.

Accordingly, for n = 1, y = sin x.

 

For n = 2, y = sin² x.

 

For n = 3, y = sin³ x.

 

We assert that 

Let our assertion be true for n = k.

i.e., 

 

Thus, our assertion is true for n = k  1.

Hence, by mathematical induction,

 

Question 63:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): 

 

ANSWER :      

 

By quotient rule,

 

 

Question 64:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): 

 

ANSWER :      

Let

By quotient rule,

 

By first principle,

 

From (i) and (ii), we obtain

 

 

Question 65:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): x⁴ (5 sin x – 3 cos x)

 

ANSWER :      

Let

By product rule,

 

 

Question 66:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (x² + 1) cos x

 

ANSWER :      

Let

By product rule,

 

 

Question 67:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (ax² + sin x) (p  + q cos x)

 

ANSWER :      

Let

By product rule,

 

 

Question 68:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): 

 

ANSWER :      

Let

By product rule,

 

Let . Accordingly,

By first principle,

 

Therefore, from (i) and (ii), we obtain

 

 

Question 69:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): 

 

ANSWER :      

Let

By quotient rule,

 

 

Question 70:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

 

ANSWER :      

Let

By quotient rule,

 ]

 

Question 71:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): 

 

ANSWER :      

Let

 

By first principle,

 

From (i) and (ii), we obtain

 

 

Question 72:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (x + sec x) (x – tan x)

 

ANSWER :      

Let

By product rule,

 

 

From (i), (ii), and (iii), we obtain

 

 

Question 73:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): 

 

ANSWER :      

Let

By quotient rule,

 

It can be easily shown that 

Therefore,