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

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

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

Limits and Derivatives Exercise 12b Solutions

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

I.

Question 1.
Find the derivative of x² – 2 at x = 10.
Solution:
Let f(x) = x² – 2, Then
f'(10) = \(\lim _{h \rightarrow 0} \frac{f(10+h)-f(10)}{h}\)

Thus, the derivative of x² – 2 at x = 10 is 20.

Question 2.
Find the derivative of x at x = 1.
Solution:
Let f(x) = x
Accordingly,

Thus, the derivative of x at x = 1 is 1.

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

Thus, the derivative of 99x at x = 100 is 99.

Question 4.
For some constants a and b, find the derivative of
(i) (x – a) (x – b)
(ii) (ax² + b)²
(iii) \(\frac{x-a}{x-b}\)
Solution:
(i) f(x) = (x – a) (x – b)
By prdduct rule, f'(x) = u’v + uv’
Let u = x – a and v = x – b
∴ f'(x) = 1(x – b) + (x – a)1
= (x – b) + (x – a)
= 2x – (a + b)

(ii) f(x) = (ax² + b)²
f'(x) = 2(ax² + b) \(\frac{\mathrm{d}}{\mathrm{dx}}\)(ax² + b)
= 2(ax² + b) (2ax)
= 4ax(ax² + b)

Question 5.
Find the derivative of \(\frac{x^n-a^n}{x-a}\) for some constant a.
Solution:
f(x) = \(\frac{x^n-a^n}{x-a}\)
By the quotient rule,

Question 6.
Find the derivative of
(i) 2x – \(\frac {3}{4}\)
(ii) (5x³ + 3x – 1) (x – 1)
(iii) x^-3(5 + 3x)
(iv) x⁵(3 – 6x^-9)
(v) x^-4 (3 – 4x^-5)
(vi) \(\frac{2}{x+1}-\frac{x^2}{3 x-1}\)
Solution:

Question 7.
Find the derivative of the following functions:
(i) sin x cos x
(ii) sec x
(iii) 5 sec x + 4 cos x
(iv) cosec x
(v) 3 cot x + 5 cosec x
(vi) 5 sin x – 6 cos x + 7
(vii) 2 tan x – 7 sec x
Solution:
(i) Let f(x) = sin x cos x, then
f'(x) = sin x \(\frac{\mathrm{d}}{\mathrm{dx}}\) (cos x) + (cos x) \(\frac{\mathrm{d}}{\mathrm{dx}}\) (sin x)
= sin x (-sin x) + cos x cos x
= cos²x – sin²x
= cos 2x

Question 8.
Find the derivative of 5 sin x + e^x log x.
Solution:

Question 9.
Find the derivative of 5^x log x + x³ e^x.
Solution:

Question 10.
If f(x) = 1 + x + x² + ………. + x¹⁰⁰, then find f'(1).
Solution:
f(x) = 1 + x + x² + ……… + x¹⁰⁰
f'(x) = 1 + 2x + 3x² + …. + 100 x⁹⁹
f'(1) = 1 + 2 + 3 + ….. + 100
= \(\frac{100 \times 101}{2}\)
= 5050

Question 11.
If f(x) = 2x² + 3x – 5, then prove that f'(0) + 3 . f'(-1) = 0.
Solution:
f(x) = 2x² + 3x – 5
⇒ f1(x) = 4x + 3
f'(0) + 3f'(-1) = 3 + 3(-4 + 3)
= 3 – 3
= 0

II.

Question 1.
Find the derivative of the following functions from the first principle.
(i) x³ – 27
(ii) (x – 1) (x – 2)
(iii) \(\frac{1}{x^2}\)
(iv) \(\frac{x+1}{x-1}\)
Solution:

Question 2.
For the function f(x) = \(\frac{x^{100}}{100}+\frac{x^{99}}{99}+\ldots . .+\frac{x^2}{2}+x+1\)
Prove that f'(1) = 100 f'(0).
Solution:

Now f'(0) = 1
and f'(1) = [1 + 1 + …. + 1] (100 times)
= 1 × 100
= 100
Thus f'(1) = 100 × f'(0)

Question 3.
Find the derivative of xⁿ + ax^n-1 + a²x^n-2 + …. + a^n-1x + aⁿ for some fixed real number a.
Solution:
Let f(x) = xⁿ + ax^n-1 + a²x^n-2 + ………. + a^n-1x + aⁿ then

Question 4.
Find the derivative of cos x from the first principle.
Solution:
Let f(x) = cos x
Then f(x + h) = cos(x + h)

Question 5.
Find the derivatives of the following functions from the first principle.
(i) \(\sqrt{x+1}\)
(ii) sin 2x
(iii) cos ax
(iv) sec 3x
(v) x sin x
(vi) cos²x
Solution: