Question

Differentiating a Logarithmic Function In Exercise, find the derivative of the function.

Answer 1

Answer:

$y' = \frac{4x}{3(x^{2} - 2)}$

Step-by-step explanation:

The derivative of $(f(x))^{n}$ is $f'(x)*n*(f(x))^{n-1}$. The derivative of $(20x^{2} + 3)^{10}$ is $400x*(20x^{2} + 3)^{9}$, for example.

We also have that the derivative of

$y = \ln{f(x)}$

Is

$y' = \frac{f'(x)}{f(x)}$

In this problem, we have that:

$y = \ln{(x^{2}-2)^{\frac{2}{3}}}$

So

$f(x) = (x^{2} - 2)^{\frac{2}{3}}$

$f'(x) = \frac{2*2x}{3}*(x^{2} - 2)^{-\frac{1}{3}}$

The derivative of the function is:

$y' = \frac{f'(x)}{f(x)}$

$y' = \frac{\frac{2*2x}{3}*(x^{2} - 2)^{-\frac{1}{3}}}{(x^{2}-2)^{\frac{2}{3}}}$

$y' = \frac{4x}{3(x^{2} - 2)}$