Derivative of sec x

Derivative of sec x

How to solve the derivative of sec x problems: formula, proof, example, and its solution.

Formula

[sec x]' = sec x tan x

The derivative of sec x is sec x tan x.

Proof

Derivative of sec x: Proof of the Formula

sec x = 1/(cos x)

Trigonometric ratio - secant

Reciprocal rule in differentiation

Derivative of cos x

1/(cos x) = sec x

Trigonometric ratio - secant

And (sin x)/(cos x) = tan x.

Trigonometric ratio - tangent

So 1/(cos x) ⋅ (sin x)/(cos x)
= sec x tan x.

So [sec x]' = sec x tan x.

Example

Find the derivative of the given function. y = sec^3 x

y = (sec x)3

So y' is equal to,
the derivative of the outer part, 3(sec x)2
times, the derivative of the inner part, (sec x tan x).

Chain rule in differentiation

Power rule in differentiation (Part 1)