Solved Examples on Derivative of Cos x

Example 1: Find the derivative of cos 4x.

Solution:

y’ = (d/dx) [cos 4x]

Applying chain rule

y’ = (d/dx) [cos 4x].(d/dx) (4x)

⇒ y’ = (-sin 4x)4

⇒ y’ = -4sin 4x

Example 2: Evaluate the derivative f(x) = (x³ + 5x² + 2x + 7) cosx.

Solution:

f(x) = (x³ + 5x² + 2x + 7)cos x

⇒ f'(x) = (d /dx)[(x³ + 5x² + 2x + 7) cosx]

Applying product rule

f'(x) = (d /dx)[(x³ + 5x² + 2x + 7)] cosx + (x³ + 5x² + 2x + 7) (d /dx)[cosx]

⇒ f'(x) = (3x² + 10x + 2)cosx – (x³ + 5x² + 2x + 7)sinx

Example 3: Use the derivative of cos x to determine the derivative of cos(cos x).

Solution:

The derivative of cos x is -sin x. By chain rule,

d(cos(cos x))/dx = -sin(cos x) . -sin x

⇒ d(cos(cos x))/dx = -sin(cos x) . -sin x

Example 4: Find the derivative of p(x) = (4x² + 9)/cosx.

Solution:

p'(x) = (d /dx)[(4x² + 9)/cosx]

By quotient rule,

p'(x) = [(d /dx)(4x² + 9) cos x – (4x² + 9)(d /dx)cosx]/ cos²x

⇒ p'(x) = [8x cosx + (4x² + 9) sinx]/ cos²x

Example 5: Find derivative of cos^-1 x.

Solution:

(d /dx) [cos^-1 x] = -1/√[1 – x²] [From Formula]

Derivative of Cosine Function, cos(x), with respect to x is -sin x. Derivative of Cos x is the change in the cosine function with respect to the variable x and represents its slope at any point x. Thus, in other words, we can say that the slope of cos x is – sin x for all real values x.

In this article, we will learn about the derivative of Cos x and its formula including the proof of the formula using the first principle of derivatives, quotient rule, and chain rule. Solved Problems and FAQs are also provided in the end along with some practice questions to learn the topic more clearly.