Derivative of Sec θ
2026-02-28 01:42 Diff
https://brightchamps.com/en-us/math/calculus/derivative-of-sec-theta

We can derive the derivative of sec θ using proofs.

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

There are several methods for proving this, such as:

By First Principle

Using Chain Rule

Using Product Rule

We will now demonstrate that the differentiation of sec θ results in sec θ tan θ using these methods:

By First Principle

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

To find the derivative of sec θ using the first principle, we will consider f(θ) = sec θ. Its derivative can be expressed as the following limit.

f'(θ) = limₕ→₀ [f(θ + h) - f(θ)] / h … (1)

Given that f(θ) = sec θ, we write f(θ + h) = sec(θ + h).

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

Using the identity cos A - cos B = -2 sin((A + B)/2) sin((A - B)/2), f'(θ) = limₕ→₀ [-2 sin((2θ + h)/2) sin(h/2)] / [h cos(θ) cos(θ + h)] = limₕ→₀ [-2 sin(θ + h/2) sin(h/2) / h] 1/[cos(θ) cos(θ + h)]

Using limit formulas, limₕ→₀ (sin h/2)/(h/2) = 1. f'(θ) = [sin θ / cos² θ] = sec θ tan θ.

Hence, proved.

Using Chain Rule

To prove the differentiation of sec θ using the chain rule, We use the formula: sec θ = 1/cos θ Let f(θ) = 1/u, where u = cos θ

Using the chain rule: d/dθ [1/u] = -1/u² · du/dθ

Let’s substitute u = cos θ in the equation, d/dθ (sec θ) = -1/(cos θ)² · (-sin θ) = sin θ/(cos θ)² Since sec θ = 1/cos θ, we write: d/dθ(sec θ) = sec θ tan θ

Using Product Rule

We will now prove the derivative of sec θ using the product rule.

The step-by-step process is demonstrated below:

Here, we use the formula, sec θ = (cos θ)⁻¹ Given that, u = 1 and v = (cos θ)⁻¹

Using the product rule formula: d/dθ [u.v] = u'v + uv' u' = d/dθ (1) = 0 v' = d/dθ ((cos θ)⁻¹) = sin θ/(cos θ)²

Again, using the product rule formula: d/dθ (sec θ) = 0 · v + 1 · v'

Substitute u = 1, u' = 0, v = (cos θ)⁻¹, and v' = sin θ/(cos θ)² d/dθ (sec θ) = sin θ/(cos θ)²

Thus: d/dθ (sec θ) = sec θ tan θ.