# RS Aggarwal Class 12 Differentiation Solutions

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Here we have provided RS Aggarwal Class 12 Differentiation Solutions for each exercise 10a, 10b, 10c, 10d, 10e, 10f, 10g, 10h, 10i, and 10j. Students from CBSE Board or ICSE Boards are strictly recommended to practice these questions to achieve higher marks in the 12 Board examination.

NCERT PDF Books For Class 12

## RS Aggarwal Class 12 Differentiation Solutions 10 A

1. Differentiate each of the following w.r.t. x: sin 4x.

Let, y = sin 4x and u = 4x

therefore, y = sin u

Differentiating above equation w.r.t. x,

………….

= cos 4x . 4

= 4 cos 4x

2. Differentiate each of the following w.r.t. x: cos 5x

Let, y = cos 5x and u = 5x

therefore, y= cos u

Differentiating above equation w.r.t. x,

………… By chain rule

………….

= – sin 5x . 5

= – 5 sin 5x

3. Differentiate each of the following w.r.t. x: tan 3x

Let, y = tan 3x and u = 3x

therefore, y= tan u

Differentiating above equation w.r.t. x,

………… By chain rule

………….

= sec2 3x . 3

= 3 sec2 3x

4. Differentiate each of the following w.r.t. x: cos x3

Let, y = cos x3 and u = x3

therefore, y= cos u

Differentiating above equation w.r.t. x,

………… By chain rule

………….

= – sin x3 . 3x2

= – 3x2 sin x3

5. Differentiate each of the following w.r.t. x: cot2x

Let, y = cot2 x and u = cot x

therefore, y= u2

Differentiating above equation w.r.t. x,

………… By chain rule

………….

= 2 cot x .(- cosec2 x)

= – 2cot x . cosec2 x

6. Differentiate each of the following w.r.t. x: tan3x

Let, y = tan3 x and u = tan x

therefore, y= u3

Differentiating above equation w.r.t. x,

………… By chain rule

………….

= 3 tan2 x .(sec2 x)

= 3 tan2 x . sec2 x

7. Differentiate each of the following w.r.t. x: cot √x

Let, y= cot √x and u = √x

therefore, y= cot u

Differentiating above equation w.r.t. x,

………… By chain rule

………….

8. Differentiate each of the following w.r.t. x:

Let, y = and u = tan x

therefore,

Differentiating above equation w.r.t. x,

………… By chain rule

………….

9. Differentiate each of the following w.r.t. x: (5 + 7x)6

Let, y = (5+7x)6 and u = (5+7x)

therefore, y = u6

Differentiating above equation w.r.t. x,

………… By chain rule

………….

= 6. (5+7x)5. (0+7) ………….

= 42. (5+7x)5

10. Differentiate each of the following w.r.t. x: (3 – 4x )5

Let, y = (3-4x)5 and u = (3-4x)

therefore, y = u5

Differentiating above equation w.r.t. x,

………… By chain rule

………….

= 5. (3-4x)4. (0-4) ………….

= -20 (3 – 4x)4

11. Differentiate each of the following w.r.t. x: (2x2 – 3x +4)5

Let, y = (2x2 – 3x + 4)5 and u = (2x2 – 3x + 4)

therefore, y = u5

Differentiating above equation w.r.t. x,

………… By chain rule

………….

= 5. (2x2 – 3x + 4)4. (4x-3+0) ………….

= 5. (2x2 – 3x + 4)4 (4x-3)

12. Differentiate each of the following w.r.t. x: (ax2 + bx + c)6

Let, y = (ax2 + bx + c)6 and u = (ax2 + bx + c)

therefore, y = u6

Differentiating above equation w.r.t. x,

………… By chain rule

………….

= 6. (ax2 + bx + c)5. (2ax+b+0) ………….

13. Differentiate each of the following w.r.t. x:

Let,

and Let, u = (x2-3x+5)3

Therefore,

For u = (x2-3x+5)3

Let, v = (x2-3x+5)

Therefore, u = (v)3

Therefore,

Differentiating above equation w.r.t. x,

………… By chain rule

………….

………….

14. Differentiate each of the following w.r.t. x:

Let,

and

Differentiating above equation w.r.t. x,

………… By chain rule

………….

………….

15. Differentiate each of the following w.r.t. x:

Let,

Multiplying numerator and denominator by (1+sin x),

……………. (1 – sin2x = cos2x)

y = sec x + tan x

Differentiating above equation w.r.t. x,

………….

16. Differentiate each of the following w.r.t. x: cos2

Let, y = cos2 x3 and u = x3

therefore, y= cos2 u

let, v = cos u

therefore, y= v2

Differentiating above equation w.r.t. x,

………… By chain rule

………….

………….

17. Differentiate each of the following w.r.t. x: sec3 (x2+1)

Let, y = sec3 (x2+1) and u = x2+1

therefore, y= sec3 u

let, v = sec u

therefore, y= v3

Differentiating above equation w.r.t. x,

………… By chain rule

………….

18. Differentiate each of the following w.r.t. x:

Let,

and u = 3x

therefore,

let, v = cos u

therefore,

Differentiating above equation w.r.t. x,

………… By chain rule

………….

19. Differentiate each of the following w.r.t. x:

Let,

and u = 2x

therefore,

let, v = sin u

therefore,

Differentiating above equation w.r.t. x,

………… By chain rule

………….

20. Differentiate each of the following w.r.t. x:

Let,

and u = 1+ cot x

therefore,

Differentiating above equation w.r.t. x,

………… By chain rule

………….

21. Differentiate each of the following w.r.t. x:

Let,

and

therefore, y= cosec3 u

let, v = cosec u

therefore, y= v3

Differentiating above equation w.r.t. x,

………… By chain rule

………….

22. Differentiate each of the following w.r.t. x:

Let,

and u = x3

therefore,

let, v = sin u

therefore,

Differentiating above equation w.r.t. x,

………… By chain rule

………….

23. Differentiate each of the following w.r.t. x:

Let,

and u = x. sin x

therefore,

Differentiating above equation w.r.t. x,

………… By chain rule

………….

………….

24. Differentiate each of the following w.r.t. x:

Let,

And

therefore,

let, v = cot u

therefore,

Differentiating above equation w.r.t. x,

………… By chain rule

………….

25. Differentiate each of the following w.r.t. x: cot3

Let, y = cot3 x2 and u = x2

therefore, y= cot3 u

let, v = cot u

therefore, y= v3

Differentiating above equation w.r.t. x,

………… By chain rule

………….

26. Differentiate each of the following w.r.t. x:

Let,

and u = ax + b

therefore,

let,

therefore,

let,

therefore,

Differentiating above equation w.r.t. x,

………… By chain rule

………….

27. Differentiate each of the following w.r.t. x:

Let,

and u = x3 + 1

therefore,

let,

therefore,

Differentiating above equation w.r.t. x,

………… By chain rule

………….

………….

28. Differentiate each of the following w.r.t. x: sin 5x. cos 3x

Let,

y = sin 5x. cos 3x

………….

Differentiating above equation w.r.t. x,

………….

………….

29. Differentiate each of the following w.r.t. x: sin 2x. sin x

Let,

y = sin 2x. sin x

………….

Differentiating above equation w.r.t. x,

………….

………….

30. Differentiate each of the following w.r.t. x: cos 4x. cos 2x

Let,

y = cos 4x. cos 2x

………….

Differentiating above equation w.r.t. x,

………….

………….

= -3 sin 6x – sin 2x

= – (3 sin 6x + sin 2x)

31. Find dy/dx, when:

Given,

Put x = tan a

Therefore, …………… eq (1)

y = sin (cos 2a) ………….

Differentiating above equation w.r.t. a ,

………….

………….

………….

But, x = tan a

…………………….. eq (2)

Now,

………… By chain rule

…………….. from eq (1) & eq (2)

…..…….

…..…….

32. Find dy/dx, when:

Given,

Differentiating above equation w.r.t. x ,

………….

………….

33. If………

Given,

Dividing numerator and denominator by cosx,

………….

………….

Differentiating above equation w.r.t. x ,

………….

………….

………….

Now,

………….

= 0

Hence Proved.

34. If……

Given,

Dividing numerator and denominator by cosx,

………….

………….

Differentiating above equation w.r.t. x ,

………….

………….

………….

Hence Proved.

## RS Aggarwal Class 12 Differentiation Solutions 10 B

1 Differentiate each of the following w.r.t. x:
(i)
(ii)
(iii)

(i) Let y = e4x z = 4x

Formula :

According to the chain rule of differentiation

(ii) Let y = e-5x z = -5x

Formula :

According to the chain rule of differentiation

(iii) Let y = z = x3

Formula :

According to the chain rule of differentiation

2. Differentiate each of the following w.r.t. x:

(i) y = z
(ii)
(iii)

(i) Let y = z = 2/x

Formula :

According to chain rule of differentiation

(ii) Let y = z =

Formula :

According to chain rule of differentiation

(iii) Let y = z = -2

Formula :

According to chain rule of differentiation

3. Differentiate each of the following w.r.t. x:
(i)
(ii)
(iii)

(i) Let y = z =

Formula :

According to chain rule of differentiation

(ii) Let y = z =

Formula :

According to chain rule of differentiation

(iii) Let y = z =

Formula :

According to chain rule of differentiation

4. Differentiate each of the following w.r.t. x:
(i) tan (log x)
(ii) log (sec x)
(iii) log (sin (x/2))

(i) Let y =tan(log x) z = log x

Formula :

According to chain rule of differentiation

(ii) Let y = log (sec x) z = sec x

Formula :

According to chain rule of differentiation

(iii) Let y = log (sin (x/2)) z = sin (x/2)

Formula :

According to chain rule of differentiation

5. Differentiate each of the following w.r.t. x:
(i)log3 x
(ii)
(iii)

(i) Let y =

Formula :

Therefore y =

According to the chain rule of differentiation

(ii) Let y =  z = -x

Formula :

According to the chain rule of differentiation

(iii) Let y =  z = x

Therefore Y =

Formula :

According to the chain rule of differentiation

6. Differentiate each of the following w.r.t. x:
(i)
(ii) log (sin (3x))
(iii)

(i) Let y =  z =

Formula :

According to chain rule of differentiation

(ii) Let y = log (sin (3x)) z = sin (3x)

Formula :

According to chain rule of differentiation

(iii) Let y =  z =

Formula :

According to the chain rule of differentiation

7. Differentiate each of the following w.r.t. x:

Let y = , z =  and w = log (x)

Formula :

According to the product rule of differentiation

8. Differentiate each of the following w.r.t. x:

Let y = , z =

Formula :

According to the chain rule of differentiation

9. Differentiate each of the following w.r.t. x:
e2x sin 3x

Let y = e2x sin 3x, z = e2x and w = sin 3x

Formula :

According to product rule of differentiation

10. Differentiate each of the following w.r.t. x:
e3x cos 2x

Let y = e3x cos 2x, z = e3x and w = cos 2x

Formula :

According to the product rule of differentiation

11. Differentiate each of the following w.r.t. x:
e-5x cot 4x

Let y = e-5x cot 4x , z = e-5x and w = cot 4x

Formula :

According to product rule of differentiation

12. Differentiate each of the following w.r.t. x:
ex log (sin 2x)

Let y = ex log (sin 2x), z = ex and w = log (sin 2x)

Formula :

According to the product rule of differentiation

13. Differentiate each of the following w.r.t. x:

Let y = , z =

Formula :

According to the chain rule of differentiation

= cosec x

14. Differentiate each of the following w.r.t. x:

Let y = , z =

Formula :

According to the chain rule of differentiation

15. Differentiate each of the following w.r.t. x:

Let y =  , u = , v = , z=

Formula :

According to the quotient rule of differentiation

If z =

According to the chain rule of differentiation

16. Differentiate each of the following w.r.t. x:

Let y =  , u = , v =

Formula :

According to the quotient rule of differentiation

If y =

()

17. Differentiate each of the following w.r.t. x:

Let y = , z = x and w =

Formula :

According to the product rule of differentiation

18. Differentiate each of the following w.r.t. x:

Let y = , z =  and w =

Formula :

According to the product rule of differentiation

19. Differentiate each of the following w.r.t. x:

Let y =  , z =  and w =

Formula :

According to the product rule of differentiation

20. Differentiate each of the following w.r.t. x:

Let y =  , u = , v =

Formula:

According to the quotient rule of differentiation

If y =

21. Differentiate each of the following w.r.t. x:

Let y = , z = x3 and w =

Formula :

According to the product rule of differentiation

22. Differentiate each of the following w.r.t. x:

Let y = , z = xcos x

Formula :

(Using product rule)

According to the chain rule of differentiation

## RS Aggarwal Class 12 Differentiation Solutions 10 C

1 Differentiate each of the following w.r.t. x:

Formulae :

i)

ii)

Let,

and u = 2x

therefore,

Differentiating above equation w.r.t. x,

…………. By chain rule

………….

2. Differentiate each of the following w.r.t. x:

Formulae :

i)

ii)

Let,

and u = x2

therefore,

Differentiating above equation w.r.t. x,

…………. By chain rule

………….

3. Differentiate each of the following w.r.t. x:

Formulae :

i)

ii)

Let,

and

therefore,

Differentiating above equation w.r.t. x,

…………. By chain rule

………….

4. Differentiate each of the following w.r.t. x:

Formulae :

i)

ii)

Let,

and

therefore,

Differentiating above equation w.r.t. x,

…………. By chain rule

………….

5. Differentiate each of the following w.r.t. x:

Formulae :

i)

ii)

Let,

and u = log x

therefore,

Differentiating above equation w.r.t. x,

…………. By chain rule

………….

6. Differentiate each of the following w.r.t. x:

Formulae :

i)

ii)

Let,

and u = ex

therefore,

Differentiating above equation w.r.t. x,

…………. By chain rule

………….

7. Differentiate each of the following w.r.t. x:

Formulae :

i)

ii)

Let,

and u = tan-1x

therefore,

Differentiating above equation w.r.t. x,

…………. By chain rule

………….

8. Differentiate each of the following w.r.t. x:

Formulae :

i)

ii)

Let,

and u = x3

therefore,

Differentiating above equation w.r.t. x,

…………. By chain rule

………….

9. Differentiate each of the following w.r.t. x:

Formulae :

i)

ii)

iii)

Let,

and

therefore,

Differentiating above equation w.r.t. x,

…………. By chain rule

………….

………

10. Differentiate each of the following w.r.t. x:

Formulae :

i)

ii)

iii)

iv)

v)

Let,

Let, u = (1+x2) and v=tan-1x

therefore, y=u.v

………

………….

………….

11. Differentiate each of the following w.r.t. x:

Formulae :

i)

ii)

iii)

Let,

and u = cot x

therefore,

Differentiating above equation w.r.t. x,

…………. By chain rule

………….

………

= -1

12. Differentiate each of the following w.r.t. x:

Formulae :

i)

ii)

iii)

Let,

and u = x4

therefore,

let,

therefore, y= log v

Differentiating above equation w.r.t. x,

………… By chain rule

………….

13. Differentiate each of the following w.r.t. x:

Formulae :

i)

ii)

Let,

and u = x2

therefore,

let,

therefore, y= v3

Differentiating above equation w.r.t. x,

………… By chain rule

………….

14. Differentiate each of the following w.r.t. x:

Formulae :

i)

ii)

iii)

Let,

and

therefore,

let,

therefore,

Differentiating above equation w.r.t. x,

………… By chain rule

………

15. Differentiate each of the following w.r.t. x:

Formulae :

i)

ii)

Let,

and u = sin-1x

therefore,

Differentiating above equation w.r.t. x,

…………. By chain rule

………….

16. Differentiate each of the following w.r.t. x:

Formulae :

i)

ii)

iii)

Let,

and

therefore,

let,

therefore,

Differentiating above equation w.r.t. x,

………… By chain rule

………

17. Differentiate each of the following w.r.t. x:

Formulae :

i)

ii)

iii)

Let,

and

therefore,

let,

therefore,

Differentiating above equation w.r.t. x,

………… By chain rule

………

18. If prove that.

Given :

To Prove :

Formulae :

i)

ii)

iii)

iv)

v)

vi)

Given equation,

Let

Therefore, y = s + t ………eq(1)

I. For

let

therefore,

Differentiating above equation w.r.t. x,

…………. By chain rule

………….

………

………eq(2)

II. For

let

, therefore,

Differentiating above equation w.r.t. x,

…………. By chain rule

………….

………

………eq(2)

Differentiating eq(1) w.r.t. x,

………

= -1 -1 ………from eq(2) and eq(3)

= -2

Hence proved !!!

19. Prove that .

To Prove :

Formulae :

i)

ii)

iii)

iv)

v)

vi)

vii)

Let,

Let

Therefore, y = s – t ………eq(1)

I. For

let

therefore,

Differentiating above equation w.r.t. x,

………

………….

………eq(2)

II. For

let

, therefore,

Differentiating above equation w.r.t. x,

…………. By chain rule

………

………

………eq(3)

Differentiating eq(1) w.r.t. x,

………

………from eq(2) and eq(3)

= 2 tan-1x

Hence proved !!!

## RS Aggarwal Class 12 Differentiation Solutions 10 D

1 Differentiate each of the following w.r.t x:

To find: Value of

The formula used: (i)

We have,

⇒

⇒

⇒

⇒

Now, we can see that

Now differentiating

2. Differentiate each of the following w.r.t x:

To find: Value of

The formula used: (i)

(ii) 1 + cos  = 2

We have,

⇒

⇒

⇒

⇒

Now, we can see that

Now differentiating

3. Differentiate each of the following w.r.t x:

To find: Value of

The formula used: (i)

(ii) 1 + cos  = 2

We have,

⇒

⇒

⇒

⇒

Now, we can see that

Now differentiating

4. Differentiate each of the following w.r.t x:

To find: Value of

The formula used: (i)

(ii) 1 + cos  = 2

We have,

⇒

⇒

⇒

Now, we can see that

Now differentiating

5. Differentiate each of the following w.r.t x:

To find: Value of

The formula used: (i)

We have,

Dividing numerator and denominator by cosx

Now, we can see that

Now differentiating

⇒ 0 + 1

⇒ 1

Ans) 1

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