Limits and Derivatives (Ex – 13.2)
Question 1.
Find the derivative of x2 – 2 at x = 10.
Solution:
let f(x) = x2 – 2
Differentiating (i) with respect to x, we get
f'(x) = 2x
At x = 10, f'(10) = 2(10) = 20.
Question 2.
Find the derivative of 99x at x = 10.
Solution:
let f(x) = 99x
Differentiating (i) with respect to x, we get
f'(x) = 90
At x = 100, f'(100) = 99.
Question 3.
Find the derivative of x at x = 10.
Solution:
let f(x) = x
Differentiating (i) with respect to x, we get
f'(x) = 1
At x = 1, f'(1) = 1.
Question 4.
Find the derivative of the following functions from first principle.
(i) x3 – 27
(ii) (x – 1)(x – 2)
(iii) 1×2
(iv) x+1/x−1
Solution:
Question 5.
For the function
Prove that f'(1) = 100f'(0)
Solution:
We have
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
Differentiating (i) with respect to x, we get
f'(x) = nxn-1 + (n – 1)axn-2 + …… + an-1
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
Question 8.
Find the derivative (xn−an)/(x−a) for some constant a.
Solution:
Let f(x) = (xn−an)/(x−a) ….(i), where a is a constant.
Differentiating (i) with respect to x, we get
Question 9.
Find the derivative oo
Solution:
(i) Let f(x) = 2x−3/4 …(1)
Differentiating (i) with respect to x, we get
f'(x) = 2·1 – 0 ⇒ f'(x) = 2.
(ii) Let f(x) = 5x3 + 3x – 1)(x – 1)
Question 10.
Find the derivative of cos x from first principle.
Solution:
Let f(x) = cos x
Question 11.
Find the derivative of the following functions:
(i) sin x cos x
(ii) secx
(iii) 5 secx + 4 cosx
(iv) cosecx
(v) 3 cotx + 5 cosecx
(vi) 5sinx – 6 cosx + 7
(vii) 2 tanx – 7 secx.
Solution:
(i) Let f(x) = sin x cos x … (1)
Differentiating (1) with respect to x, we get
