I am having problems figuring out how to derive this.

Question

I have the function

$$\tag{1} f(x)=\ln\sqrt{8+\cos^2x}$$

So we derive it as follows:

$$\tag{2} f(x)=\ln(8+\cos^2x)^\frac{1}{2}$$

$$\tag{3} f(x)=\frac{1}{2}\ln(8+\cos^2x)$$

$$\tag{4} f'(x)=\frac{1}{2}\left[\frac{-2 \cos x{\sin x}}{8+\cos^2x}\right]$$

$$\tag{5} f'(x)=\left[\frac{-\cos x{\sin x}}{8+\cos ^2x}\right]$$

I'm having problems with everything after step 2, a step by step explanation would be greatly appreciated. Thanks. For example where does the sin come from and why is there a cos in the numerator now. A detailed process of beginning to end of solving this (what laws are used and why etc..) would be greatly appreciated.

Here is the image of the website and how it goes about doing the steps, I thought it's be better for me to try and type it out unless anybody is curious. !

Answer 1

Recall that $\frac{d}{dx}\ln(x) = \frac1x$. So using the chain rule, $$\frac{d}{dx}\ln(f(x)) = \frac{f'(x)}{f(x)}$$

Do you think you can work it out from there?

Answer 2

You just have to use the Chain Rule a couple of times.

The derivative of $$(1/2)\ln(8+\cos^2 x)$$

is $$(1/2)(1/(8+\cos^2 x))(2\cos x(-\sin x)).$$