Differentiate the function with respect to x.  cos x³.sin² (x⁵)

Solution:

Let f(x) = cos x³.sin²(x⁵)

⇒ d/dx [cos x³.sin²(x⁵)]

If a function is lying inside the given function then we say the function is composite.

Since the given function in the problem is a composite function.

It can be solved by using the chain rule of derivatives.

i.e we need to differentiate all the functions present in the problem separately and then multiply at the end.

By using chain rule of derivative.

we get

= sin²(x⁵) × d/dx (cos x³) + cos x³ × d/dx [sin²(x⁵)]

= sin²(x⁵) × (− sin x³) × d/dx (x³) + cos x³ × 2sin (x⁵).d/dx [sin x⁵]

= −sin x³sin² (x⁵) × 3x² + 2sin x⁵cos x³.cos x⁵ × d/dx (x⁵)

= −3x²sin x³.sin² (x⁵) + 2sin x⁵cos x⁵cos x³ × 5x⁴

⇒ d/dx [cos x³.sin²(x⁵)] = 10x⁴ sin x⁵ cos x⁵cos x³ − 3x² sin x³sin² (x⁵)

NCERT Solutions Class 12 Maths - Chapter 5 Exercise 5.2 Question 6

Differentiate the function with respect to x. cos x³.sin²(x⁵)

Summary:

By using chain rule of derivative, we get  the derivative of the function with respect to x. cos x³.sin²(x⁵) is d/dx [cos x³.sin²(x⁵)] = 10x⁴ sin x⁵ cos x⁵cos x³ − 3x² sin x³sin² (x⁵)