# Find the derivative of the following. g(t) = ln \sqrt[3]{6t^2-1}

## Question:

Find the derivative of the following.

{eq}g(t) = ln \sqrt[3]{6t^2-1} {/eq}

## Derivative:

In this exercise, we apply differentiation rules to calculate derivative of a function.

1. derivative rule for a composed functions.

{eq}\frac{d f(g(x))}{dx} = f'(g(x)g'(x) {/eq}

2. derivative of logarithm function

{eq}\frac{d \ln f(x)}{dx} = \frac{1}{f(x)}f'(x) {/eq}

Given the function

{eq}g(t) = \ln \sqrt[3]{6t^2-1} = \ln (6t^2-1)^{1/3} = \frac{1}{3} \ln (6t^2-1) {/eq}

its derivative is calculated as

{eq}g'(t) = \frac{1}{3} D[ \ln (6t^2-1)] D[6t^2-1] = \\ = \frac{1}{3(6t^2-1)} 12t = \\ = \frac{4t}{6t^2-1} {/eq}