write f' in terms of g'

[Sophie]

Question: Write f' in terms of g' in each case. Simplify your answers.


\(\displaystyle \L\\\begin{array}{l}
f(x) = x^3 g(e^{2x} ) \\
g(x) = e^{2x} \\
g'(x) = 2e^{2x} \\
f'(x) = \frac{d}{{dx}}x^3 g + x^3 \frac{d}{{dx}}g \\
f'(x) = 3x^2 e^{2x} + x^3 e^{2x} 2 \\
f'(x) = x^2 e^{2x} \left( {3 + 2x} \right) \\
\end{array}\)

My problem is I think I have gone of on my own little tangent. I would appreciate some direction if this is wrong.

Thanks, Sophie

 

[soroban]

Hello, Sophie!

Could you state the original problem?
As written, the problem is quite awful . . .

Write \(\displaystyle f'(x)\) in terms of \(\displaystyle g'(x)\). .Simplify your answers.

\(\displaystyle f(x) \:= \:x^3\cdot g(e^{2x})\;\) . . . g of e^2x ?

\(\displaystyle g(x) \:= \:e^{2x}\;\) . . . are sure this is true?

Then: \(\displaystyle \L\:g\left(e^{2x}\right) \:=\:e^{(e^{^{2x}})}\)

And: \(\displaystyle \L\:f(x)\:=\:x^3\cdot e^{(e^{^{2x}})}\;\) . . . which looks overly complicated

 

[Sophie]

The origonal question is:

Write f' in terms of g' in each case. Simplify your answers.

a.
\(\displaystyle f(x) = x^3 g(e^{2x} )\)

There is nothing else I am afraid.

I am not sure the following is correct as I am grasping at straws in an attempt to answer the question!
\(\displaystyle g(x) = e^{2x}\)


Thanks Sophie