# Find Derivative of ln u | Log u Derivative

The derivative of ln u is equal to 1/u du/dx, and this is the derivative of ln(u) with respect to x. Here we differentiate the natural logarithm of u with respect to x.

Recall that, ln u = logeu.

## Derivative of ln(u)

If u is a function of x, then the derivative of ln u is provided below.

Explanation:

Let us put

z = ln u.

⇒ ez = u

Now differentiating both sides with respect to x, we get that

$e^z \dfrac{dz}{dx} = \dfrac{du}{dx}$

⇒ $e^{\ln u} \dfrac{dz}{dx} = \dfrac{du}{dx}$ as z=lnu.

⇒ $u \dfrac{dz}{dx} = \dfrac{du}{dx}$, using the formula elnx = x.

⇒ $\dfrac{dz}{dx} = \dfrac{1}{u} \dfrac{du}{dx}$.

⇒ $\dfrac{d}{dx} (\ln u) = \dfrac{1}{u} \dfrac{du}{dx}$.

So the derivative of lnu (natural logarithm of u) with respect to x is equal to 1/u du/dx.

## What is the Derivative of log u?

As the derivative of logax is equal to 1/(x lna), using the same argument as above we conclude that the derivative of logau with respect to x is equal to 1/(u lna) du/dx. That is,

$\dfrac{d}{dx}(\log_a u)=\dfrac{1}{u \ln a} \dfrac{du}{dx}$.