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RD Chapter 30 Derivatives Ex 30.1 Solutions

Question - 1 : - Find the derivative of f(x) = 3x at x = 2

Answer - 1 : -

Given:
f(x) = 3x
By using the derivative formula,

Question - 2 : -

Find the derivativeof f(x) = x2 – 2 at x = 10

Answer - 2 : -

Given:

f(x) = x2 – 2

By using the derivative formula,

= 0 + 20 = 20

Hence,

Derivative of f(x) = x2 –2 at x = 10 is 20

Question - 3 : - Find the derivative of f(x) = 99x at x = 100.

Answer - 3 : -

Given:
f(x) = 99x
By using the derivative formula,

Question - 4 : - Find the derivative of f(x) = x at x = 1

Answer - 4 : -

Given:
f(x) = x
By using the derivative formula,

Question - 5 : - Find the derivative of f(x) = cos x at x = 0

Answer - 5 : -

Solution:
Given:
f(x) = cos x
By using the derivative formula,

Question - 6 : - Find the derivative of f(x) = tan x at x = 0

Answer - 6 : -

Given:
f(x) = tan x
By using the derivative formula,

Question - 7 : -
Find the derivatives of the following functions at the indicated points:
(i) sin x at x = π/2
(ii) x at x = 1
(iii) 2 cos x at x = π/2
(iv) sin 2xat x = π/2

Answer - 7 : -

(i) sin x at x = π/2
Given:
f (x) = sin x
By using the derivative formula,

[Since it is ofindeterminate form. Let us try to evaluate the limit.]

We know that 1 – cos x = 2 sin2(x/2)

(ii) x at x = 1

Given:

f (x) = x

By using the derivative formula,

(iii) 2 cos x at x = π/2
Given:
f (x) = 2 cos x
By using the derivative formula,
(iv) sin 2xat x = π/2
Solution:
Given:
f (x) = sin 2x
By using the derivative formula,
[Since it is of indeterminate form. We shall apply sandwich theorem to evaluate the limit.]
Now, multiply numerator and denominator by 2, we get

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