Derivative of Cosine x
2026-02-28 08:33 Diff
https://brightchamps.com/en-us/math/calculus/derivative-of-cosine-x

We can derive the derivative of cos x using proofs.

To show this, we will use the trigonometric identities along with the rules of differentiation.

There are several methods we use to prove this, such as:

By First Principle

Using Chain Rule

Using Product Rule

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

By First Principle

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

To find the derivative of cos x using the first principle, we will consider f(x) = cos x.

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

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

We now use the formula for cosine subtraction: cos A - cos B = -2sin((A + B)/2)sin((A - B)/2). f'(x) = limₕ→₀ [ -2sin((x + h + x)/2)sin((x + h - x)/2) ] / h = limₕ→₀ [ -2sin(x + h/2)sin(h/2) ] / h = limₕ→₀ -sin(x) [ sin(h/2) / (h/2) ] · 1/2

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

Hence, proved.

Using Chain Rule

To prove the differentiation of cos x using the chain rule,

We use the formula: cos x = 1/sin x Consider f(x) = 1 and g(x) = sin x

So we get, cos x = f(x)/g(x) By quotient rule: d/dx [f(x) / g(x)] = [f'(x) g(x) - f(x) g'(x)] / [g(x)]²… (1)

Let’s substitute f(x) = 1 and g(x) = sin x in equation (1), d/dx (cos x) = [(0) (sin x) - (1) (cos x)] / (sin x)² = -cos x / sin² x …(2)

Here, we use the identity sin² x + cos² x = 1.

Substituting this into (2), d/dx (cos x) = -sin x. Since sin x = sin x, we write: d/dx(cos x) = -sin x

Using Product Rule

We will now prove the derivative of cos x using the product rule.

The step-by-step process is demonstrated below:

Here, we use the formula, cos x = 1/sin x cos x = (1) · (sin x)⁻¹

Given that, u = 1 and v = (sin x)⁻¹

Using the product rule formula: d/dx [u.v] = u'. v + u. v' u' = d/dx (1) = 0 (substitute u = 1)

Here we use the chain rule: v = (sin x)⁻¹ = (sin x)⁻¹ (substitute v = (sin x)⁻¹) v' = -1 · (sin)⁻² · d/dx (sin x) v' = -cos x / sin² x Again, use the product rule formula: d/dx (cos x) = u'. v + u. v'

Let’s substitute u = 1, u' = 0, v = (sin x)⁻¹, and v' = -cos x / sin² x

When we simplify each term: We get, d/dx (cos x) = -sin x

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