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  • RBSE Class 11

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

June 7, 2019 by Fazal Leave a Comment

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

Question 2.
Find the derivative of 49x at x = 50.
Solution:
Let f(x) = 49x
RBSE Solutions for Class 11 Maths Chapter 10 Limits and Derivatives

Question 3.
Find the derivative of the following function from first principle:
RBSE Solutions for Class 11 Maths Chapter 10 Limits and Derivatives
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
RBSE Solutions for Class 11 Maths Chapter 10 Limits and Derivatives

(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
RBSE Solutions for Class 11 Maths Chapter 10 Limits and Derivatives
RBSE Solutions for Class 11 Maths Chapter 10 Limits and Derivatives
RBSE Solutions for Class 11 Maths Chapter 10 Limits and Derivatives
RBSE Solutions for Class 11 Maths Chapter 10 Limits and Derivatives

Question 4.
For the function
RBSE Solutions for Class 11 Maths Chapter 10 Limits and Derivatives
Prove that f'(1) = 100 f'(0).
Solution:
RBSE Solutions for Class 11 Maths Chapter 10 Limits and Derivatives
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 + axn – 1 + a2xn – 2 + … + an – 1 x + an
Solution:
Let y =f(x) = xn + axn – 1 + a2xn – 2 + …… + an – 1x + an
Then, derivative of f(x),
RBSE Solutions for Class 11 Maths Chapter 10 Limits and Derivatives

Question 6.
For some constant a and b, find the derivative of the following functions :
RBSE Solutions for Class 11 Maths Chapter 10 Limits and Derivatives
Solution:
(i) Let y = f(x) = (x – a) (x – b) or y = f(x) = x2 – (a + b)x + ab
Then, derivative of given function
RBSE Solutions for Class 11 Maths Chapter 10 Limits and Derivatives
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
RBSE Solutions for Class 11 Maths Chapter 10 Limits and Derivatives
RBSE Solutions for Class 11 Maths Chapter 10 Limits and Derivatives
= 4a2x3 + 4abx = 4ax(ax2 + b)
Hence, derivative of given function (ax2 + b2)2
= 4a2x3 + 4abx or 4ax(ax2 + b)

RBSE Solutions for Class 11 Maths Chapter 10 Limits and Derivatives
We know that if any function is in the form of fraction, then its derivative
RBSE Solutions for Class 11 Maths Chapter 10 Limits and Derivatives

Question 7.
For any constant a, find the derivative of
RBSE Solutions for Class 11 Maths Chapter 10 Limits and Derivatives.
Solution:
RBSE Solutions for Class 11 Maths Chapter 10 Limits and Derivatives
RBSE Solutions for Class 11 Maths Chapter 10 Limits and Derivatives

Question 8.
Find the derivative of the following 3
RBSE Solutions for Class 11 Maths Chapter 10 Limits and Derivatives
Solution:
RBSE Solutions for Class 11 Maths Chapter 10 Limits and Derivatives
RBSE Solutions for Class 11 Maths Chapter 10 Limits and Derivatives
(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
RBSE Solutions for Class 11 Maths Chapter 10 Limits and Derivatives
= 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
RBSE Solutions for Class 11 Maths Chapter 10 Limits and Derivatives
Hence, derivative of given function
RBSE Solutions for Class 11 Maths Chapter 10 Limits and Derivatives
We can also solve this equation by product rule of derivative.
RBSE Solutions for Class 11 Maths Chapter 10 Limits and Derivatives

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
RBSE Solutions for Class 11 Maths Chapter 10 Limits and Derivatives

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.
RBSE Solutions for Class 11 Maths Chapter 10 Limits and Derivatives
= – 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

(ii) Let f(x) = sec x
RBSE Solutions for Class 11 Maths Chapter 10 Limits and Derivatives
Hence, derivative of the given function sec x = sec x tan x

(iii) Let f(x) = cosec x
Then, derivative of f(x)
RBSE Solutions for Class 11 Maths Chapter 10 Limits and Derivatives
= – cosec x cot x
Hence, derivative of the given function cosec x
= – cosec x cot x

(iv) Let f(x) = 3 cot x + 5 cosec x
RBSE Solutions for Class 11 Maths Chapter 10 Limits and Derivatives
Hence, derivative of the given function 3 cot x + 5 cosec x is – 3 cosec2 x – 5 cosec x cot x

(v) Let f(x) = 5 sin x – 6 cos x + 7
RBSE Solutions for Class 11 Maths Chapter 10 Limits and Derivatives
Hence, derivative of the given function 5 sin x – 6 cos x + 7 is 5 cos x + 6 sin x.

RBSE Solutions for Class 11 Maths

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