Differentiate cos (sin x) with respect to x:

Given:

f(x) = cos(sin(x))

Concept used:

The chain rule states that if you have a composition of functions, like f(g(x)), the derivative is f′(g(x)) × g′(x).

Calculation:

⇒ f(u) = cos(u)

⇒ f'(u) = -sin(u) (derivative of cos(u))

⇒ g(x) = sin(x)

⇒ g'(x) = cos(x) (derivative of sin(x))

⇒ \(\dfrac{d}{dx}\)[cos(sin(x))] = f'(g(x)) × g'(x)

⇒ \(\dfrac{d}{dx}\)[cos(sin(x))] = -sin(sin(x)) × cos(x)

∴ The derivative of cos(sin(x)) with respect to x is −cos x ⋅ sin (sin x)