** Chapter 1 Differentiation Ex 1.2 **

Chapter 1 Differentiation Ex 1.2

**Question 1.Find the derivative of the function y = f (x) using the derivative of the inverse function x = ( y) in the following(i) y = âˆšxSolution:**

y = âˆšx â€¦ (1)

We have to find the inverse function of y = f(x), i.e. x in terms of y.

From (1),

= x âˆ´ x = y

^{2}

We have to find the inverse function of y = f(x), i.e. x in terms of y.

From (1),

We have to find the inverse function of y = f(x), i.e. x in terms of y.

From (1),

**(iv) y = log (2x â€“ 1)Solution:y = log (2x â€“ 1) â€¦(1)**

We have to find the inverse function of y = f(x), i.e. x in terms of y.

From (1),

**(v) y = 2x + 3Solution:y = 2x + 3 â€¦.(1)**

We have to find the inverse function of y = f(x), i.e. x in terms of y.

From (1),

**(vi) y = â€“ 3Solution:y = â€“ 3 â€¦.(1)**

We have to find the inverse function of y = f(x), i.e. x in terms of y.

From (1),

= y + 3

âˆ´ x = log(y + 3)

âˆ´ x = (y) = log(y + 3)

**(vii) y = Solution:y = â€¦.(1)**

We have to find the inverse function of y = f(x), i.e. x in terms of y.

From (1),

2x â€“ 3 = log y âˆ´ 2x = log y + 3

**Question 2.Find the derivative of the inverse function ofthe following(i) y = Solution:**

y =

Differentiating w.r.t. x, we get

**(ii) y = x cos xSolution:**

y = x cos x

Differentiating w.r.t. x, we get

**(iii) y = xÂ·Solution:**

y = xÂ·

Differentiating w.r.t. x, we get

**(iv) y = + logxSolution:**

y = + logx

Differentiating w.r.t. x, we get

**(v) y = x logxSolution:**

y = x logx

Differentiating w.r.t. x, we get

**Question 3.Find the derivative of the inverse of the following functions, and also fid their value at the points indicated against them.**

The derivative of inverse function of y = f(x) is given by

**(ii) y = + 3x + 2, at x = 0Solution:**

y = + 3x + 2

Differentiating w.r.t. x, we get

The derivative of inverse function of y = f(x) is given by

**(iii) y = 3 + 2 log x ^{3}, at x = 1Solution:**

y = 3 + 2 log

= 3 + 6 log x

Differentiating w.r.t. x, we get

The derivative of inverse function of y = f(x) is given by

**(iv) y = sin (x â€“ 2) + , at x = 2Solution:**

y = sin (x â€“ 2) +

Differentiating w.r.t. x, we get

**Question 4.If f(x) = + x â€“ 2, find ()â€™ (0).Question is modified.If f(x) = + x â€“ 2, find ()â€™ (-2).Solution:**

f(x) = + x â€“ 2 â€¦.(1)

Differentiating w.r.t. x, we get

**Question 5.Using derivative proveSolution:**

let f(x) =

Differentiating w.r.t. x, we get

Since, f'(x) = 0, f(x) is a constant function.

Let f(x) = k.

For any value of x, f(x) = k

Let x = 0.

Then f(0) = k â€¦.(2)

From (1), f(0) =

Since, f'(x) = 0, f(x) is a constant function.

Let f(x) = k.

For any value of x, f(x) = k, where |x| > 1

Let x = 2.

Then, f(2) = k â€¦â€¦(2)

**Question 6.Diffrentiate the following w. r. t. x.(i) ta(log x)Solution:**

Let y = ta (log x)

Differentiating w.r.t. x, we get

**(ii) Solution:**

Let y =

Differentiating w.r.t. x, we get

**(iii) Solution:**
Let y =

Differentiating w.r.t. x, we get

**(iv) Solution:**

Let y =

Differentiating w.r.t. x, we get

Differentiating w.r.t. x, we get

Differentiating w.r.t. x, we get

**(vii) Solution:Let y = **

Differentiating w.r.t. x, we get

Differentiating w.r.t. x, we get

**Question 7.**

Solution:

Solution:

Solution:

Solution:

Solution:

Solution:

Solution:

**(xi) (cosec x + cot x)Solution:**

Let y = (cosec x + cot x)

Solution:

**Question 8.(i) Solution:**

**(ii) Solution:**

**(iii) Solution:**

**(iv) Solution:**

**(v) Solution:**

= e

^{x}.

**(vi) Solution:**

**Question 9.Diffrentiate the following w. r. t. x.Solution:**

Solution:

**Solution:**

Solution:

**(v) **

**Solution:**

**Solution:**

Solution:

Solution:

**Solution:**

Solution:

**Question 10.Diffrentiate the following w. r. t. x.**