Inter 1st Year Maths Limits and Derivatives Solutions Exercise 12c

Practicing the AP Board Solutions Class 11 Maths and Chapter 12 Inter 1st Year Maths Limits and Derivatives Solutions Exercise 12c Pdf Download will help students to clear their doubts quickly.

Intermediate 1st Year Maths Limits and Derivatives Solutions Exercise 12c

Limits and Derivatives Exercise 12c Solutions

Limits and Derivatives Class 11 Exercise 12c Solutions – Limits and Derivatives 12c Exercise Solutions

I. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r, and s are fixed non-zero constants and m and n are integers):

Question 1.
x + a
Solution:
Let f(x) = x + a
Accordingly, f(x + h) = x + h + a

Question 2.
(px + q) (\(\frac {r}{x}\) + s)
Solution:
Let f(x) = (px + q) (\(\frac {r}{x}\) + s)
By Leibniz’s product rule,

Question 3.
(ax + b) (cx + d)²
Solution:
Let f(x) = (ax + b) (cx + d)²
By Leibniz’s product rule,

Question 4.
\(\frac{a x+b}{c x+d}\)
Solution:
By the quotient rule,

Question 5.
\(\frac{1+\frac{1}{x}}{1-\frac{1}{x}}\)
Solution:

Question 6.
\(\frac{1}{a x^2+b x+c}\)
Solution:
Let f(x) = \(\frac{1}{a x^2+b x+c}\)
By the quotient rule,

Question 7.
\(\frac{a x+b}{p x^2+q x+r}\)
Solution:
Let f(x) = \(\frac{a x+b}{p x^2+q x+r}\)
By the quotient rule,

Question 8.
\(\frac{p x^2+q x+r}{a x+b}\)
Solution:

Question 9.
\(\frac{a}{x^4}-\frac{b}{x^2}\) + cos x
Solution:
Let f(x) = \(\frac{a}{x^4}-\frac{b}{x^2}\) + cos x

Question 10.
4√x – 2
Solution:
Let f(x) = 4√x – 2

Question 11.
(ax + b)ⁿ
Solution:
Let f(x) = (ax + b)ⁿ
Accordingly, f(x + h) = {a(x + h) + b}ⁿ = (ax + ah + b)ⁿ

Question 12.
(ax + b)ⁿ (cx + d)^m
Solution:
Let f(x) = (ax + b)ⁿ (cx + d)^m
Then f(x + h) = (ax + ah + b)ⁿ (cx + ch + d)^m

Question 13.
sin(x + a)
Solution:
Let f(x) = sin (x + a)
Then f(x + h) = sin (x + h + a)

Question 14.
Find the derivatives of the following functions.
(i) cotⁿx
(ii) cosec⁴x
(iii) sin^mx . cosⁿx
(iv) sin mx . cos nx
(v) log (tan 5x)
(vi) \(\log \left(\frac{x^2+x+2}{x^2-x+2}\right)\)
(vii) cos(log x + ex)
Solution:

II.

Question 1.
Find the derivative of the following functions from the first principle:
(i) -x
(ii) (-x)^-1
(iii) sin(x + 1)
(iv) \(\cos \left(x-\frac{\pi}{8}\right)\)
Solution:

Find the derivatives of the following functions (it is to be understood that a, b, c, d, p, q, r, and s are fixed non-zero constants and in and n are integers):

Question 2.
cosec x cot x
Solution:
Let f(x) = cosec x cot x
Then f'(x) = cosec x (cot x)’ + cot x (cosec x)’
= cosec x (-cosec²x) + cot x (-cosec x . cot x)’
= -cosec³x – cosec x cot²x

Question 3.
\(\frac{\cos x}{1+\sin x}\)
Solution:

Question 4.
\(\frac{\sin x+\cos x}{\sin x-\cos x}\)
Solution:

Question 5.
\(\frac{\sec x-1}{\sec x+1}\)
Solution:

Question 6.
sinⁿx
Solution:

Question 7.
\(\frac{a+b \sin x}{c+d \cos x}\)
Solution:

Question 8.
\(\frac{\sin (x+a)}{\cos x}\)
Solution:

Question 9.
x⁴ (5 sin x – 3 cos x)
Solution:
Let f(x) = x⁴ (5 sin x – 3 cos x)
By the product rule,
f'(x) = x⁴ \(\frac{\mathrm{d}}{\mathrm{dx}}\)(5 sin x – 3 cos x) + (5 sin x – 3 cos x) \(\frac{\mathrm{d}}{\mathrm{dx}}\)(x4)
= \(x^4\left[5 \frac{d}{d x}(\sin x)-3 \frac{d}{d x}(\cos x)\right]+(5 \sin x-3 \cos x) \frac{d}{d x}\left(x^4\right)\)
= x⁴ [5 cos x – 3(-sin x)] + (5 sin x – 3 cos x) (4x³)
= x³ [5x cos x + 3x sin x + 20 sin x – 12 cos x]

Question 10.
(x² + 1) cos x
Solution:
Let f(x) = (x² + 1) cos x
By the product rule,
f'(x) = (x² + 1) \(\frac{\mathrm{d}}{\mathrm{dx}}\)(cos x) + cos x \(\frac{\mathrm{d}}{\mathrm{dx}}\)(x² + 1)
= (x² + 1) (-sin x) + cos x (2x)
= -x² sin x – sin x + 2x cos x

Question 11.
(ax² + sin x) (p + q cos x)
Solution:
Let f(x) = (ax² + sin x) (p + q cos x)
By the product rule,
f'(x) = (ax² + sin x) \(\frac{\mathrm{d}}{\mathrm{dx}}\)(p + q cos x) + (p + q cos x) \(\frac{\mathrm{d}}{\mathrm{dx}}\)(ax2 + sin x)
= (ax² + sin x) (-q sin x) + (p + q cos x) (2ax + cos x)
= -q sin x (ax² + sin x) + (p + q cos x) (2ax + cos x)

Question 12.
(x + cos x) (x – tan x)
Solution:
Let f(x) = (x + cos x) (x – tan x)
By the product rule,
f'(x) = (x + cos x) \(\frac{\mathrm{d}}{\mathrm{dx}}\)(x – tan x) + (x – tan x) \(\frac{\mathrm{d}}{\mathrm{dx}}\)(x + cos x)
= (x + cos x) (1 – sec²x) + (x – tan x) (1 – sin x)
= (x + cos x) (-tan²x) + (x – tan x) (1 – sin x)

Question 13.
\(\frac{4 x+5 \sin x}{3 x+7 \cos x}\)
Solution:

Question 14.
\(\frac{x^2 \cos \left(\frac{\pi}{4}\right)}{\sin x}\)
Solution:

Question 15.
\(\frac{x}{1+\tan x}\)
Solution:

Question 16.
(x + sec x) (x – tan x)
Solution:
\(\frac{\mathrm{d}}{\mathrm{dx}}\)[(x + sec x) (x – tanx)]
= (x + sec x) \(\frac{\mathrm{d}}{\mathrm{dx}}\)(x – tan x) + (x – tan x) \(\frac{\mathrm{d}}{\mathrm{dx}}\)(x + sec x)
= (x + sec x) (1 – sec²x) + (x – tan x) [1 + sec x tan x]
= (x + sec x) (-tan²x) + (x – tan x) (1 + sec x tan x)

Question 17.
\(\frac{x}{\sin ^n x}\)
Solution: