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)

= x² cos² x − x cos x sin x − x² sin² x − x cos x sin x

= x² (cos² x − sin² x) − 2x cos x sin x

= x² cos 2x − x sin 2x

= x (x cos 2x − sin 2x)

Question 12. Differentiate (x sin x + cos x) (e^x + x² log x) with respect to x.

Solution:

We have,

=> y = (x sin x + cos x) (e^x + 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) (e^x + x² 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 sec² x − 8 sin x sec² x + 4 cos x − 8 tan x cos x

= 

= −10 sec² x − 8 tan x sec x + 4 cos x − 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) (−sinx)

= 2x cos x − 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 e^x cos x with respect to x.

Solution:

We have, 

=> y = x³ e^x 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 − 3 cos x) with respect to x.

Solution:

We have,

=> y = x⁴ (5 sin x − 3 cos x)

On differentiating both sides, we get,

On using product rule we get,

= 

= 

= 20 x³ sin x − 12 x³ cos x + 5x⁴ cos x + 3x⁴ sin x