Question

Find f′(x) for f(x) = cos^2(3x^3).

Answer 1

\(\large\color{black}{ \displaystyle f(x)=\left(\cos~3x^3~\right)^2 }\)

Answer 2

Tell me what is the derivative of:

■ cos(x)
■ 3x\(^3\)

Answer 3

cosx is -sinx
and 3x^3 is 9x^2

Answer 4

yes, so you are to use the chain rule twice:

\(\large\color{black}{ \displaystyle f(x)=\left(\cos~3x^3~\right)^2 }\)

\(\large\color{black}{ \displaystyle f'(x)=\color{red}{2}\left(\cos~3x^3~\right)^{2\color{red}{-1}} \color{blue}{\times (-\sin~3x^3)}\color{green}{\times (9x^2)} }\)

Answer 5

In red I denoted how the power rule is applied, then in blue is the first chain rule for the inner function of the cosine, and then in green I denoted the chain rule for the angle of 3x\(^3\).

Answer 6

oh thank you that makes so much more sense! So that's it right?

Answer 7

Yes, everything is right... you just need to simplify the final answer if that is required to any extent.

Answer 8

Basically, what happened is:

\(\large\color{black}{ \displaystyle \frac{d}{dx} f\left(\color{blue}{g\left(\color{green}{h\left(x\right)}\right)}\right)= f'\left(\color{blue}{g\left(\color{green}{h\left(x\right)}\right)}\right)\times \color{blue}{g'\left(\color{green}{h\left(x\right)}\right)}\times \color{blue}{\color{green}{h'\left(x\right)}}}\)

Answer 9

I'm also having a hard time with this equation any chance you can help me on this one too?
Find f′(x) for f(x) = ln(x^2 + e^3x).

Answer 10

How much further would you have to condense that final answer you gave me

Answer 11

The derivative of \(\color{black}{\ln x}\) is \(\color{black}{1/x}\),
thus the derivative of \(\color{black}{\ln \left(f(x)\right)}\) is \(\color{black}{\cfrac{1}{ \left(f(x)\right)}\times f'(x)}\)

Answer 12

I will also help you by telling that:

\(\large\color{black}{ \displaystyle \frac{d}{dx} \left[e^{f(x)}\right]=f'(x) \cdot e^{f(x)} }\)

Answer 13

The derivative of x\(^2\), I bet you already know, and so is the derivative of e\(^{3x}\)

G\(\large ☼☼\)D LUCK \(\color{darkgoldenrod }{\Huge ☻}\)