Derivative of loga x

Derivative of log_a x

How to solve the derivative of loga x problems: formula, proof, examples, and their solutions.

Formula

[log_a x]' = 1/(x ln a)

The derivative of loga x is 1/(x ln a).

Proof

Derivative of log_a x: Proof of the Formula

Change of base formula

Take the constant 1/(ln a) out.

Derivative of ln x

So [loga x]' = 1/(x ln a).

Example 1

Find the derivative of the given function. y = log_2 (x^3 - 8x)

y = log2 (x3 - 8x)

So y' is equal to,
the derivative of the outer part, 1/[(x3 - 8x) ln 2]
times, the derivative of the inner part, (3x2 - 8).

Chain rule in differentiation

Derivative of polynomials

Example 2

Find the derivative of the given function. y = log |x^2 - 1|

y = log |x2 - 1|
= log10 |x2 - 1|

Common logarithms

So y' is equal to,
the derivative of the outer part, 1/[(x2 - 1) ln 10]
times, the derivative of the inner part, 2x.

Chain rule in differentiation

Derivative of polynomials