RBSE Solutions for Class 11 Maths Chapter 10 Limits and Derivatives Ex 10.3

Rajasthan Board RBSE Class 11 Maths Chapter 10 Limits and Derivatives Ex 10.3

Question 1.
Find the derivative of x² – 2 at x = 10.
Solution:
Let f(x) = x² – 2

Question 2.
Find the derivative of 49x at x = 50.
Solution:
Let f(x) = 49x

Question 3.
Find the derivative of the following function from first principle:

Solution:
(i) Let y = x³ – 16
Again, let y + δy = (x + δx)³ – 16
⇒ δy = (x + δx)³ – 16 – y
⇒ δy = (x + δx)³ – 16 – x³ + 16
⇒ δy = (x + δx)³ – x³

(ii) Let y = (x – 1) (x – 2) = x² – 3x + 2
Again, let y + δy = (x + δx)2 – 3(x + δx) + 2
⇒ δy = (x + δx)² – 3(x + δx) + 2 – x² + 3x – 2

Question 4.
For the function

Prove that f'(1) = 100 f'(0).
Solution:

Then, putting 1 and 0 in place of x.
f'(1)= (1⁹⁹ + 1⁹⁸ + … + 1)+ 1
= 1 + 1 + 1 + …+ 99 term + 1
= 99+ 1 = 100 and f'(0) = 1
Hence, f'(1)= 100
∵ f'(1) = 100 f'(0) Hence Proved.

Question 5.
For any constant real number a, find the derivative of:
xⁿ + ax^{n – 1} + a²x^{n – 2} + … + a^{n – 1} x + aⁿ
Solution:
Let y =f(x) = xⁿ + ax^{n – 1} + a²x^{n – 2} + …… + a^{n – 1}x + aⁿ
Then, derivative of f(x),

Question 6.
For some constant a and b, find the derivative of the following functions :

Solution:
(i) Let y = f(x) = (x – a) (x – b) or y = f(x) = x² – (a + b)x + ab
Then, derivative of given function

Hence, derivative of given function (x – a) (x – b)
= 2x – a – b
(ii) Let y = f(x) = (ax² + b)²
or y = f(x) = a²x⁴+ 2abx² + b²
Then, derivative of given function

= 4a²x³ + 4abx = 4ax(ax² + b)
Hence, derivative of given function (ax² + b²)²
= 4a²x³ + 4abx or 4ax(ax² + b)

We know that if any function is in the form of fraction, then its derivative

Question 7.
For any constant a, find the derivative of
.
Solution:

Question 8.
Find the derivative of the following 3

Solution:

(ii) Let y = f(x) = (5x³ + 3x – 1) (x – 1)
The given function is product of two function.
Then, derivative of product of two functions

= 20x³ – 15x² + 6x – 4
Hence, derivative of given function = 20x³ – 15x² + 6x – 4.

(iii) Let y = x⁵(3 – 6x^-9)
Then, derivative of given function

Hence, derivative of given function

We can also solve this equation by product rule of derivative.

Question 9.
Find the derivative of cos x by first principle.
Solution:
Let
f(x) = cos x, then f(x + h) = cos(x + h)
Then

Question 10.
Find the derivatives of the following :
(i) sin x cos x
(ii) sec x
(iii) cosec x
(iv) 3 cot x + 5 cosec x
(v) 5 sin x – 6 cos x + 7
Solution:
(i) Let f(x) = sin x. cos x, which is product of two functions.
So, formula of derivative of product of two functions.

= – sin² x + cos² x
= cos² x – sin² x
= cos 2x ( ∵ cos² x – sin² x = cos²x)
Hence, derivative of given function sin x cos x = cos 2x