# Derivative of log_{a} *x*

How to solve the derivative of log_{a} *x* problems: formula, proof, examples, and their solutions.

## Formula

The derivative of log_{a} *x* is 1/(*x* ln *a*).

## Proof

Take the constant 1/(ln a) out.

So [log_{a} *x*]' = 1/(*x* ln *a*).

## Example 1

*y* = log_{2} (*x*^{3} - 8*x*)

So *y*' is equal to,

the derivative of the outer part, 1/[(*x*^{3} - 8*x*) ln 2]

times, the derivative of the inner part, (3*x*^{2} - 8).

Chain rule in differentiation

Derivative of polynomials

## Example 2

*y* = log |*x*^{2} - 1|

= log_{10} |*x*^{2} - 1|

Common logarithms

So *y*' is equal to,

the derivative of the outer part, 1/[(*x*^{2} - 1) ln 10]

times, the derivative of the inner part, 2*x*.

Chain rule in differentiation

Derivative of polynomials