OP Malhotra Class 11 Maths Solutions Chapter 19 Differentiation Ex 19(a)

The availability of ISC OP Malhotra Solutions Class 11 Chapter 19 Differentiation Ex 19(a) encourages students to tackle difficult exercises.

S Chand Class 11 ICSE Maths Solutions Chapter 19 Differentiation Ex 19(a)

Question 1.
2x
Solution:
Let y = 2x …(1)
Let δx be the increment in x and corresponding increment in y be δy
∴ y + δy = 2 (x + δx) …(2)
subtracting eqn. (1) from eqn. (2); we get
δy = 2δx ; On dividing both sides by δx
∴ \(\frac{\delta y}{\delta x}\) = 2, Taking limits as δx → 0
Thus, \(\underset{\delta x \rightarrow 0}{\mathrm{Lt}}\) \(\frac{\delta y}{\delta x}\) = \(\frac{d y}{d x}\) = \(\underset{\delta x \rightarrow 0}{\mathrm{Lt}}\) = 2 = 2
∴ \(\frac{d }{d x}\)(2x) = 2

Question 2.
(x – 1)²
Solution:
Let y = f(x) = (x – 1)²
∴ f(x + δx) = (x + δx – 1)²
Thus by first principle, we have

Question 3.
x³
Solution:
Let y = f(x) = x³
∴ f (x + δx) = (x + δx)³
Then by first principle, we have

Question 4.
\(\frac{1}{\sqrt{x}}\)
Solution:

Question 5.
\(\sqrt{x+1}\); x > – 1
Solution:
Let y = f(x) = \(\sqrt{x+1}\) ∴ f(x + δx) = \(\sqrt{x+\delta x+1}\)
Then by first principle, we have

Question 6.
\(\frac{2 x+3}{3 x+2}\)
Solution:
Let y = f(x) = \(\frac{2 x+3}{3 x+2}\)
∴ f(x + δx) = \(\frac{2(x+\delta x)+3}{3(x+\delta x)+2}\)
Then by first principle, we have

Question 7.
\(\frac{1}{\sqrt{x+a}}\)
Solution:
Given y = f(x) = \(\frac{1}{\sqrt{x+a}}\)
∴f(x + δx) = \(\frac{1}{\sqrt{x+\delta x+a}}\)
Then by first principle, we have

Question 8.
x + \(\frac{1}{x}\)
Solution:
Let y = f(x) = x + \(\frac{1}{x}\)
∴f(x + δx) = (x + δx) + \(\frac{1}{x+δx}\)
Then by first principle, we have

Question 9.
\(\frac{1}{\sqrt{2 x+3}}\)
Solution:
Let y = f(x) = \(\frac{1}{\sqrt{2 x+3}}\)
∴ f (x + δx) = \(\frac{1}{\sqrt{2(x+\delta x)+3}}\)
Then by first principle, we have

Question 10.
\(\frac{1}{x^{\frac{3}{2}}}\)
Solution:
Let y = f(x) = \(\frac{1}{x^{\frac{3}{2}}}\)
∴ f(x + δx) = \(\frac{1}{(x+\delta x)^{3 / 2}}\)
Then by first principle, we have

Question 11.
(x + 1) (2x – 3)
Solution:
Let y = f(x) = (x + 1) (2x – 3)
∴f(x + δx) = (x + δx + 1) (2x – 3 + 2 δx)
Then by first principle, we have

Question 12.
\(\frac{x^2+1}{x}\)
Solution:
Let y = f(x) = \(\frac{x^2+1}{x}\) = x + \(\frac { 1 }{ x }\)
∴f(x + δx) = (x + δx) + \(\frac{1}{x+\delta x}\)
Then by first principle, we have