Question

Find f′(x) for f(x) = ln(x^2 + e^3x).

Answer 1

-12A'328

Answer 2

JK

Answer 3

You are going to have to use the Chain Rule.

You know that in a case,
\(\large\color{black}{ \displaystyle f(x)=\ln x }\)

the derivative is going to be,
\(\large\color{black}{ \displaystyle \frac{d}{dx}\left [\ln x \right]=\frac{1}{x} }\)

And this way, (by applying the Chain Rule,) in a case,
\(\large\color{black}{ \displaystyle f(x)=\ln({~}\color{red}{f(x){~}}) }\)

the derivative is going to be,
\(\large\color{black}{ \displaystyle \frac{d}{dx}\left [ \ln({~}\color{red}{f(x){~}}) \right]=\frac{1}{{}\color{red}{f(x){}}} \times \color{red}{f'(x)} }\)

Answer 4

And therefore when you have:
\(\large\color{black}{ \displaystyle f(x)= \ln(x^2 + e^{3x}) }\)

you know that your derivative is going to be:

\(\large\color{black}{ \displaystyle f'(x)= \frac{1}{x^2 + e^{3x}}\times \left[\frac{d}{dx}~~x^2 + e^{3x}\right] }\)