# NCERT solution class 12 chapter 5 Continuity and Differentiability exercise 5.3 mathematics part 1

## EXERCISE 5.3

#### Question 1:

Find

:

The given relationship is

Differentiating this relationship with respect to x, we obtain

#### Question 2:

Find

:

The given relationship is

Differentiating this relationship with respect to x, we obtain

#### Question 3:

Find

:

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

:

The given relationship is

Differentiating this relationship with respect to x, we obtain

#### Question 5:

Find

:

The given relationship is

Differentiating this relationship with respect to x, we obtain

[Derivative of constant function is 0]

#### Question 6:

Find

:

The given relationship is

Differentiating this relationship with respect to x, we obtain

#### Question 7:

Find

:

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

:

The given relationship is

Differentiating this relationship with respect to x, we obtain

#### Question 9:

Find  :

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

:

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  :

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

:

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

:

The given relationship is

Differentiating this relationship with respect to x, we obtain

#### Question 14:

Find

:

The given relationship is

Differentiating this relationship with respect to x, we obtain

Find

: