## NCERT Solutions for Class 12 Maths Chapter 5 – Continuity and Differentiability Ex 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
: