Derivative: Chain Rule with two natural logarithms

[ReneG]

I'm a freshman in highschool trying to get a headstart into calculus, so please bear with my limited knowledge.

Anyways, I have \(\displaystyle \frac{d}{dx} \ln{(\ln{(2 - \cos{x})})}\)

I'm assuming I have to use the Chain Rule, but I have no idea. If someone could show me how to solve this problem step by step, It'll be greatly appreciated. Thanks.

 

[soroban]

Hello, ReneG!

I'm a freshman in high school trying to get a headstart into calculus. . Good for you!

\(\displaystyle \text{Differentiate: }\:y \:=\:\ln\big[\ln(2-\cos x)\big]\)

You are right . . . This requires the Chain Rule.

\(\displaystyle \dfrac{dy}{dx} \;=\;\underbrace{\dfrac{1}{\ln(2-\cos x)}}_{\text{outer log}}\cdot \underbrace{\dfrac{1}{2-\cos x}}_{\text{inner log}} \cdot \underbrace{\sin x}_{\text{inside}} \;=\;\dfrac{\sin x}{(2-\cos x)\ln(2-\cos x)}\)

 

[ReneG]

Awesome, I see it now! I got the derivative of the outer log, but I just couldn't figure anything out after that. Thank you so much!