# Class 11 RD Sharma Solutions – Chapter 30 Derivatives – Exercise 30.4 | Set 2

Last Updated : 16 May, 2021

### Question 11. Differentiate (x sin x + cos x) (x cos x âˆ’ sin x) with respect to x.

Solution:

We have,

=> y = (x sin x + cos x) (x cos x âˆ’ sin x)

On differentiating both sides, we get,

On using product rule we get,

On using chain rule, we get,

On using product rule again, we get,

= (x cos x âˆ’ sin x) (x cos x) + (x sin x + cos x) (âˆ’x sin x)

= x2 cos2 x âˆ’ x cos x sin x âˆ’ x2 sin2 x âˆ’ x cos x sin x

= x2 (cos2 x âˆ’ sin2 x) âˆ’ 2x cos x sin x

= x2 cos 2x âˆ’ x sin 2x

= x (x cos 2x âˆ’ sin 2x)

### Question 12. Differentiate (x sin x + cos x) (ex + x2 log x) with respect to x.

Solution:

We have,

=> y = (x sin x + cos x) (ex + x2 log x)

On differentiating both sides, we get,

On using product rule we get,

On using chain rule, we get,

On using product rule again, we get,

= (x cos x) (ex + x2 log x) +(x sin x + cos x) (ex + 2x log x + x)

### Question 13. Differentiate (1 âˆ’ 2 tan x) (5 + 4 sin x) with respect to x.

Solution:

We have,

=> y = (1 âˆ’ 2 tan x) (5 + 4 sin x)

On differentiating both sides, we get,

On using product rule we get,

= âˆ’10 sec2 x âˆ’ 8 sin x sec2 x + 4 cos x âˆ’ 8 tan x cos x

= âˆ’10 sec2 x âˆ’ 8 tan x sec x + 4 cos x âˆ’ 8 sin x

### Question 14. Differentiate (1 + x2) cos x with respect to x.

Solution:

We have,

=> y = (1 + x2) cos x

On differentiating both sides, we get,

On using product rule we get,

= cos x (2x) + (1 + x2) (âˆ’sinx)

= 2x cos x âˆ’ sin x(1 + x2) (sinx)

### Question 15. Differentiate sin2 x with respect to x.

Solution:

We have,

=> y = sin2 x

=> y = (sin x) (sin x)

On differentiating both sides, we get,

On using product rule we get,

= sin x cos x + sin x cos x

= 2 sin x cos x

= sin 2x

### Question 16. Differentiate  with respect to x.

Solution:

We have,

=> y =

On differentiating both sides, we get,

= 0

### Question 17. Differentiate  with respect to x.

Solution:

We have,

=> y =

On differentiating both sides, we get,

On using product rule we get,

On using product rule again, we get,

### Question 18. Differentiate x3 ex cos x with respect to x.

Solution:

We have,

=> y = x3 ex cos x

On differentiating both sides, we get,

On using product rule we get,

On using product rule again, we get,

### Question 19. Differentiate  with respect to x.

Solution:

We have,

=> y =

=> y =

On differentiating both sides, we get,

On using product rule we get,

On using product rule again, we get,

### Question 20. Differentiate x4 (5 sin x âˆ’ 3 cos x) with respect to x.

Solution:

We have,

=> y = x4 (5 sin x âˆ’ 3 cos x)

On differentiating both sides, we get,

On using product rule we get,

= 20 x3 sin x âˆ’ 12 x3 cos x + 5x4 cos x + 3x4 sin x

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