Updated By SaraNextGen

On March 11, 2024, 11:35 AM

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