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

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


 

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