# Derivative of sec *x*

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

## Formula

The derivative of sec *x* is sec *x* tan *x*.

## Proof

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

*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)