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

**Question 11. Differentiate (x sin x + cos x) (x cos x − sin x) with respect to x.**

**Solution:**

We have,

=> y = (x sin x + cos x) (x cos x − 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 − sin x) (x cos x) + (x sin x + cos x) (−x sin x)

= x

^{2}cos^{2}x − x cos x sin x − x^{2}sin^{2}x − x cos x sin x= x

^{2}(cos^{2}x − sin^{2}x) − 2x cos x sin x= x

^{2}cos 2x − x sin 2x= x (x cos 2x − sin 2x)

**Question 12. Differentiate (x sin x + cos x) (e**^{x} + x^{2} log x) with respect to x.

^{x}+ x

^{2}log x) with respect to x.

**Solution:**

We have,

=> y = (x sin x + cos x) (e

^{x}+ x^{2}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) (e

^{x }+ x^{2 }log x) +(x sin x + cos x) (e^{x }+ 2x log x + x)

**Question 13. Differentiate (1 − 2 tan x) (5 + 4 sin x) with respect to x.**

**Solution:**

We have,

=> y = (1 − 2 tan x) (5 + 4 sin x)

On differentiating both sides, we get,

On using product rule we get,

=

=

= −10 sec

^{2}x − 8 sin x sec^{2}x + 4 cos x − 8 tan x cos x=

= −10 sec

^{2}x − 8 tan x sec x + 4 cos x − 8 sin x

**Question 14. Differentiate (1 + x**^{2}) cos x with respect to x.

^{2}) cos x with respect to x.

**Solution:**

We have,

=> y = (1 + x

^{2}) cos xOn differentiating both sides, we get,

On using product rule we get,

=

= cos x (2x) + (1 + x

^{2}) (−sinx)= 2x cos x − sin x(1 + x

^{2}) (sinx)

**Question 15. Differentiate sin**^{2 }x with respect to x.

^{2 }x with respect to x.

**Solution:**

We have,

=> y = sin

^{2}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 x**^{3} e^{x} cos x with respect to x.

^{3}e

^{x}cos x with respect to x.

**Solution:**

We have,

=> y = x

^{3 }e^{x}cos xOn 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 x**^{4} (5 sin x − 3 cos x) with respect to x.

^{4}(5 sin x − 3 cos x) with respect to x.

**Solution:**

We have,

=> y = x

^{4}(5 sin x − 3 cos x)On differentiating both sides, we get,

On using product rule we get,

=

=

= 20 x

^{3}sin x − 12 x^{3}cos x + 5x^{4}cos x + 3x^{4}sin x

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