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

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NCERT QUESTION

 

( Limits And Derivatives )

 

Question 1:Evaluate the Given limit: NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

 

ANSWER :       

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

 

Question 2:Evaluate the Given limit: NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

 

ANSWER :       

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

 

Question 3:     Evaluate the Given limit: NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

 

ANSWER :       

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

 

Question 4:     Evaluate the Given limit: NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

 

ANSWER :

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

 

Question 5:     Evaluate the Given limit: NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

 

ANSWER :       

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

 

Question 6:     Evaluate the Given limit: NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

 

ANSWER :       

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

Put x  1 = y so that y → 1 as x → 0.

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

 

Question 7:     Evaluate the Given limit: NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

 

ANSWER :       

 

At x = 2, the value of the given rational function takes the form NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs.

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

 

Question 8:     Evaluate the Given limit: NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

 

ANSWER :       

At x = 2, the value of the given rational function takes the form NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs.

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

 

Question 9:     Evaluate the Given limit: NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

 

ANSWER :       

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

 

Question 10:   Evaluate the Given limit:  NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

 

ANSWER :       

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

At z = 1, the value of the given function takes the form NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs.

Put NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs so that z →1 as x → 1.

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

 

Question 11:   Evaluate the Given limit: NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

 

ANSWER :       

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

 

Question 12:   Evaluate the Given limit: NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

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

 

ANSWER :       

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

At x = –2, the value of the given function takes the form NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs.

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

 

Question 13:   Evaluate the Given limit: NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

 

ANSWER :       

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

At x = 0, the value of the given function takes the form NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs.

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

 

Question 14:   Evaluate the Given limit: NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

 

ANSWER :       

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

At x = 0, the value of the given function takes the form NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs.

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

 

Question 15:   Evaluate the Given limit: NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

 

ANSWER :       

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

It is seen that x → π ⇒ (π – x) → 0

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

 

Question 16:   Evaluate the given limit: NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

 

 ANSWER :      NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

 

Question 17:   Evaluate the Given limit: NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

 

 ANSWER :      NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

At x = 0, the value of the given function takes the form NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs.

Now,

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

 

Question 18:   Evaluate the Given limit: NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

 ANSWER :      

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

At x = 0, the value of the given function takes the form NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs.

Now,

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

 

Question 19:   Evaluate the Given limit: NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

 

 ANSWER :      

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

 

Question 20:   Evaluate the Given limit: NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

 

 ANSWER :      

At x = 0, the value of the given function takes the form NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs.

Now,

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

 

Question 21:   Evaluate the Given limit: NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

 

 ANSWER :      At x = 0, the value of the given function takes the form NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs.

Now,

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

 

Question 22:   NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

 

ANSWER :       

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

At, NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs the value of the given function takes the form NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs.

Now, put NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs so that NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs.

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

 

Question 23:   Find NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobsf(x) and NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobsf(x), where f(x) =NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

 

ANSWER :       

The given function is

f(x) =NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

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

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

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

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

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

 

Question 24:   Find f(x), where f(x) =NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

 

ANSWER :       

The given function is

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

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

 

Question 25:   Evaluate NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobsf(x), where f(x) = NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

 

ANSWER :       

The given function is

f(x) = NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

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

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

 

Question 26:   Find NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobsf(x), where f(x) =NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

 

ANSWER :       

The given function is

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

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

 

Question 27:   Find NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobsf(x), where f(x) =NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

 

ANSWER :       

The given function is f(x) =.NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

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

 

Question 28: Suppose f(x) = NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobsand if NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobsf(x) = f(1) what are possible values of a and b?

 

ANSWER :       

The given function is

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

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

 

Question 29:   Let NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobsbe fixed real numbers and define a function

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

What is NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobsf(x)? For some NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs compute NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobsf(x).

 

ANSWER :       

The given function is NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

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

 

Question 30:   If f(x) = NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

For what value (s) of a does NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobsf(x) exists?

 

ANSWER :       

The given function is

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

When a < 0,

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

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

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

When a > 0

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

Thus,  NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs exists for all a ≠ 0.

 

Question 31:   If the function f(x) satisfies NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs, evaluate NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs.

 

ANSWER :       

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

 

Question 32:

If NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs For what integers m and n does NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs and NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobsexist?

 

ANSWER :       The given function is

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

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

Thus, NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs exists if mn.

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

 

Thus, NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs 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,

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

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,

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

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,

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

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)   NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs                                  (iv) NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

 

ANSWER :       

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

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

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

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

(iii) Let NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs. Accordingly, from the first principle,

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

(iv) Let NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs. Accordingly, from the first principle,

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

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

 

Question 37:   For the function

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

Prove that NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

 

ANSWER :       

The given function is

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

Thus, NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

 

Question 38:   Find the derivative of NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobsfor some fixed real number a.

 

ANSWER :       

Let NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

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

 

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

(i) (x – a) (x – b)                   (ii) (ax2b)2                          (iii)NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

 

ANSWER :       

 (i) Let f (x) = (x – a) (x – b)

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

(ii) Let NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

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

(iii)NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

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

By quotient rule,

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

 

Question 40:   Find the derivative of NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs for some constant a.

 

ANSWER :       

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

By quotient rule,

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

 

Question 41:   Find the derivative of

(i)  NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs                                                                   (ii) (5x3 + 3x – 1) (x – 1)

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

(v) x–4 (3 – 4x–5)                                                         (vi)NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

 

ANSWER :       

 (i) Let NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

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

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

By Leibnitz product rule,

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

(iii) Let f (x) = x– 3 (5 +3x)

By Leibnitz product rule,

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

(iv) Let f (x) = x5 (3 – 6x–9)

By Leibnitz product rule,

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

(v) Let f (x) = x–4 (3 – 4x–5)

By Leibnitz product rule,

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

(vi) Let f (x) = NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

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

By quotient rule,

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

 

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

ANSWER :       

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

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

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

 

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 

 

ANSWER :       

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

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

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

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

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

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

 

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

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

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

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

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

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

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

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

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

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

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

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

 

Question 44:

Find the derivative of the following functions from first principle:

(i) –x (ii) (–x)–1 (iii) sin (x + 1)

(iv) NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

 

ANSWER :       

 (i) Let f(x) = –x. Accordingly, NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

By first principle,

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

(ii) Let NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs. Accordingly, NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

By first principle,

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

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

(iii) Let f(x) = sin (x + 1). Accordingly, NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

By first principle,

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

 

(iv) Let NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs. Accordingly, NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

By first principle,

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

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

 

Question 45:   Find the derivative of the following functions (it is to be understood that abcdp, q, r and s are fixed non-zero constants and m and n are integers): (xa)

 

ANSWER :       

Let f(x) = x  + a. Accordingly, NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

By first principle,

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

 

Question 46:   Find the derivative of the following functions (it is to be understood that abcdp, q, r and s are fixed non-zero constants and m and n are integers): NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

 

ANSWER :       NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

By Leibnitz product rule,

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

 

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

 

ANSWER :       

Let NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

By Leibnitz product rule,

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

 

Question 48:   Find the derivative of the following functions (it is to be understood that abcdp, q, r and s are fixed non-zero constants and m and n are integers):  NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

 

ANSWER :       Let NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

By quotient rule,

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

 

Question 49:   Find the derivative of the following functions (it is to be understood that abcdp, q, r and s are fixed non-zero constants and m and n are integers):NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

 

ANSWER :       

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

By quotient rule,

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

 

Question 50:

Find the derivative of the following functions (it is to be understood that abcdp, q, r and s are fixed non-zero constants and m and n are integers): NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

 

ANSWER :       

Let NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

By quotient rule,

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

 

Question 51:

Find the derivative of the following functions (it is to be understood that abcdp, q, r and s are fixed non-zero constants and m and n are integers): NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

 

ANSWER :       

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

By quotient rule,

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

 

Question 52:

Find the derivative of the following functions (it is to be understood that abcdp, q, r and s are fixed non-zero constants and m and n are integers): NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

 

ANSWER :       

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

By quotient rule,

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

 

Question 53:

Find the derivative of the following functions (it is to be understood that abcdp, q, r and s are fixed non-zero constants and m and n are integers): NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

 

ANSWER :       

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

 

Question 54:

Find the derivative of the following functions (it is to be understood that abcdp, q, r and s are fixed non-zero constants and m and n are integers): NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

 

ANSWER :       

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

 

Question 55:

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

 

ANSWER :       

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

By first principle,

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

 

Question 56:

Find the derivative of the following functions (it is to be understood that abcdp, q, r and s are fixed non-zero constants and m and n are integers): (axb)n (cxd)m

 

ANSWER :       

Let NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

By Leibnitz product rule,

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

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

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

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

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

 

Question 57:

Find the derivative of the following functions (it is to be understood that abcdp, q, r and s are fixed non-zero constants and m and n are integers): sin (xa)

 

ANSWER :       

Let NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

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

By first principle,

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

 

Question 58:

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

 

ANSWER :       

Let NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

By Leibnitz product rule,

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

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

By first principle,

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

Now, let f2(x) = cosec x. Accordingly, NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

By first principle,

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

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

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

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

 

Question 59:

Find the derivative of the following functions (it is to be understood that abcdp, q, r and s are fixed non-zero constants and m and n are integers): NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

 

ANSWER :       

Let NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

By quotient rule,

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

 

Question 60:

Find the derivative of the following functions (it is to be understood that abcdp, q, r and s are fixed non-zero constants and m and n are integers): NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

 

ANSWER :       

Let NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

By quotient rule,

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

 

Question 61:

Find the derivative of the following functions (it is to be understood that abcdp, q, r and s are fixed non-zero constants and m and n are integers): NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

 

ANSWER :       

Let NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

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

By quotient rule,

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

 

Question 62:

Find the derivative of the following functions (it is to be understood that abcdp, 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.

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

For n = 2, y = sin2 x.

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

For n = 3, y = sin3 x.

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

We assert that NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

Let our assertion be true for nk.

i.e., NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

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

Thus, our assertion is true for nk  1.

Hence, by mathematical induction, NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

 

Question 63:

Find the derivative of the following functions (it is to be understood that abcdp, q, r and s are fixed non-zero constants and m and n are integers): NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

 

ANSWER :       

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

By quotient rule,

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

 

Question 64:

Find the derivative of the following functions (it is to be understood that abcdp, q, r and s are fixed non-zero constants and m and n are integers): NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

 

ANSWER :       

Let NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

By quotient rule,

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

By first principle,

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

From (i) and (ii), we obtain

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

 

Question 65:

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

 

ANSWER :       

Let NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

By product rule,

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

 

Question 66:

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

 

ANSWER :       

Let NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

By product rule,

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

 

Question 67:

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

 

ANSWER :       

Let NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

By product rule,

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

 

Question 68:

Find the derivative of the following functions (it is to be understood that abcdp, q, r and s are fixed non-zero constants and m and n are integers): NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

 

ANSWER :       

Let NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

By product rule,

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

Let NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs. Accordingly, NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

By first principle,

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

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

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

 

Question 69:

Find the derivative of the following functions (it is to be understood that abcdp, q, r and s are fixed non-zero constants and m and n are integers): NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

 

ANSWER :       

Let NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

By quotient rule,

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

 

Question 70:

Find the derivative of the following functions (it is to be understood that abcdp, q, r and s are fixed non-zero constants and m and n are integers): NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

 

ANSWER :       

Let NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

By quotient rule,

 NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs]

 

Question 71:

Find the derivative of the following functions (it is to be understood that abcdp, q, r and s are fixed non-zero constants and m and n are integers):  NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

 

ANSWER :       

Let NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

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

By first principle,

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

From (i) and (ii), we obtain

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

 

Question 72:

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

 

ANSWER :       

Let NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

By product rule,

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

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

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

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

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

 

Question 73:

Find the derivative of the following functions (it is to be understood that abcdp, q, r and s are fixed non-zero constants and m and n are integers): NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

 

ANSWER :       

Let NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

By quotient rule,

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

It can be easily shown that  NCERT Solutions - Limits and Derivatives, Class 11, Maths Notes - Government Jobs

Therefore,

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

 


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