## EXERCISE 5.3

#### Page No 169:

#### Question 1:

Find

:

#### Answer:

The given relationship is

Differentiating this relationship with respect to *x*, we obtain

#### Question 2:

Find

:

#### Answer:

The given relationship is

Differentiating this relationship with respect to *x*, we obtain

#### Question 3:

Find

:

#### Answer:

The given relationship is

Differentiating this relationship with respect to *x*, we obtain

Using chain rule, we obtain** **and

From (1) and (2), we obtain

#### Question 4:

Find

:

#### Answer:

The given relationship is

Differentiating this relationship with respect to *x*, we obtain

#### Question 5:

Find

:

#### Answer:

The given relationship is

Differentiating this relationship with respect to *x*, we obtain

[Derivative of constant function is 0]

#### Question 6:

Find

:

#### Answer:

The given relationship is

Differentiating this relationship with respect to *x*, we obtain

#### Question 7:

Find

:

#### Answer:

The given relationship is

Differentiating this relationship with respect to *x*, we obtain

Using chain rule, we obtain

From (1), (2), and (3), we obtain

#### Question 8:

Find

:

#### Answer:

The given relationship is

Differentiating this relationship with respect to *x*, we obtain

#### Question 9:

Find :

#### Answer:

We have,y = sin-12×1 + x2put x = tan θ ⇒ θ = tan-1xNow, y = sin-12 tan θ1 + tan2θ⇒y = sin-1sin 2θ, as sin 2θ=2 tan θ1 + tan2θ⇒y = 2θ, as sin-1sin x=x⇒y = 2 tan-1x⇒dydx = 2 × 11 + x2, because dtan-1xdx=11 + x2⇒dydx = 21 + x2

#### Question 10:

Find

:

#### Answer:

The given relationship is

It is known that,

Comparing equations (1) and (2), we obtain

Differentiating this relationship with respect to *x*, we obtain

#### Question 11:

Find :

#### Answer:

The given relationship is,

On comparing L.H.S. and R.H.S. of the above relationship, we obtain

Differentiating this relationship with respect to *x*, we obtain

sec2y2.ddxy2=ddxx

⇒sec2y2×12dydx=1

⇒dydx=2sec2y2

⇒dydx=21+tan2y2

∴

dydx=21+x2

#### Question 12:

Find

:

#### Answer:

The given relationship is

Differentiating this relationship with respect to *x*, we obtain

Using chain rule, we obtain

From (1), (2), and (3), we obtain

**Alternate method**

⇒

Differentiating this relationship with respect to *x*, we obtain

#### Question 13:

Find

:

#### Answer:

The given relationship is

Differentiating this relationship with respect to *x*, we obtain

#### Question 14:

Find

:

#### Answer:

The given relationship is

Differentiating this relationship with respect to *x*, we obtain

#### Question 15:

Find

:

#### Answer:

The given relationship is

Differentiating this relationship with respect to *x*, we obtain