Derivative of cos(5x)
2026-02-28 06:11 Diff
https://brightchamps.com/en-us/math/calculus/derivative-of-cos(5x)

We can derive the derivative of cos(5x) using proofs. To show this, we will use trigonometric identities along with the rules of differentiation. There are several methods we use to prove this, such as:

  1. By First Principle
  2. Using Chain Rule
  3. Using Product Rule

We will now demonstrate that the differentiation of cos(5x) results in -5sin(5x) using the above-mentioned methods:

By First Principle

The derivative of cos(5x) can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient.

To find the derivative of cos(5x) using the first principle, we will consider f(x) = cos(5x). Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h … (1)

Given that f(x) = cos(5x), we write f(x + h) = cos(5(x + h)).

Substituting these into equation (1), f'(x) = limₕ→₀ [cos(5(x + h)) - cos(5x)] / h = limₕ→₀ [cos(5x + 5h) - cos(5x)] / h = limₕ→₀ [-2sin(5x + 5h/2)sin(5h/2)] / h = limₕ→₀ [-2sin(5x + 5h/2)sin(5h/2)] / (h/2) · (1/2)

Using limit formulas, limₕ→₀ sin(5h/2) / (5h/2) = 1/5, f'(x) = -5sin(5x)

Hence, proved.

Using Chain Rule

To prove the differentiation of cos(5x) using the chain rule, We use the formula: Let u = 5x, then cos(u) By chain rule: d/dx [cos(u)] = -sin(u) · du/dx … (1)

Let’s substitute u = 5x in equation (1), d/dx (cos(5x)) = -sin(5x) · 5 = -5sin(5x)

Hence, proved.

Using Product Rule

We will now prove the derivative of cos(5x) using the product rule. The step-by-step process is demonstrated below: Here, we use the formula, Let u = cos(x) and v = 5, then cos(5x) = cos(x · 5)

Using the product rule formula: d/dx [u · v] = u' · v + u · v' u' = d/dx (cos(x)) = -sin(x). (substitute u = cos(x))

Here we use the chain rule: v = 5 v' = 0

Again, use the product rule formula: d/dx (cos(5x)) = (-sin(x)) · 5 + cos(x) · 0 = -5sin(x)

Thus: d/dx (cos(5x)) = -5sin(5x)