NCERT Solutions for Class 11 Maths Chapter 12 Limits and Derivatives Ex 12.2
Ex 12.2 Class 11 Maths Question 1.
Find the derivative of x2 – 2 at x = 10.
Solution:
Let f(x) = x2 – 2 …(i)
Differentiating (i) with respect to x, we get
f'(x) = 2x
At x = 10, f'(10) = 2(10) = 20.
Ex 12.2 Class 11 Maths Question 2.
Find the derivative of x at x = 1.
Solution:
Let f(x) = x …(i)
Differentiating (i) with respect to x, we get
f'(x) = 1
At x = 1, f'(1) = 1.
Ex 12.2 Class 11 Maths Question 3.
Find the derivative of 99x at x = 100.
Solution:
Let f(x) = 99x …(i)
Differentiating (i) with respect to x, we get
f'(x) = 99
At x = 100, f'(100) = 99.
Ex 12.2 Class 11 Maths Question 4.
Find the derivative of the following functions from first principle.
(i) x3 – 27
(ii) (x – 1) (x – 2)
(iii) 1/x2
(iv) (x+1)/(x−1)
Solution:
Ex 12.2 Class 11 Maths Question 5.
For the function
Prove that f'(1) = 100f'(0).
Solution:
Given,
Ex 12.2 Class 11 Maths Question 6.
Find the derivative of xn + axn-1 + a2xn-2 + …. + an-1x + an for some fixed real number a.
Solution:
Let f(x) = xn + axn-1 + a2xn-2 + …. + an-1x + an …(i)
Differentiating (i) with respect to x, we get
f'(x) = nxn-1 + (n – 1)axn-2 + …… + an-1
Ex 12.2 Class 11 Maths Question 7.
For some constants a and b, find the derivative of
(i) (x – a) (x – b)
(ii) (ax2 + b)2
(iii) (x−a)/(x−b)
Solution:
(i) Let f(x) = (x – a) (x – b) ….(1)
Differentiating (1) with respect to x, we get
f'(x) = (x – a) (x – b)’ + (x – a)’ (x – b)
⇒ f'(x) = (x – a) + (x – b) = 2x – a – b
Ex 12.2 Class 11 Maths Question 8.
Find the derivative (xn−an)/(x−a) for some constant a.
Solution:
Let f(x) = (xn−an)/(x−a), for some constant a ….(i)
Differentiating (i) with respect to x, we get
Ex 12.2 Class 11 Maths Question 9.
Find the derivative of
Solution:
(i) Let f(x) = 2x – 3/4 …(1)
Differentiating (1) with respect to x, we get
f'(x) = 2 – 0 ⇒ f'(x) = 2.
(ii) Let f(x) = (5x3 + 3x – 1)(x – 1)
Ex 12.2 Class 11 Maths Question 10.
Find the derivative of cos x from first principle.
Solution:
Let f(x) = cos x
Ex 12.2 Class 11 Maths Question 11.
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) 3 cot x + 5 cosec x
(vi) 5 sin x – 6 cos x + 7
(vii) 2 tan x – 7 sec x
Solution:
(i) Let f(x) = sin x cos x … (1)
Differentiating (1) with respect to x, we get
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NCERT Solutions for Maths Class 9