# Thread: First and second derivative of e^6e^x?

1. ## First and second derivative of e^6e^x?

derivative of e^6e^x?

2. Originally Posted by BugzLooney
derivative of e^6e^x?
Is it $\displaystyle e^{6e^x}$ ?

3. Yes that is the question indeed.

4. The first derivate, using the chain rule

$\displaystyle f(x)=e^{6e^x}$
$\displaystyle f'(x)=e^{6e^x} \cdot (6e^x)'$
$\displaystyle f'(x)=6e^{6e^x} \cdot (e^x)'$
$\displaystyle f'(x)=6e^{6e^x+x}$

5. consider the expression equal to say t. take logs on both sides of the equation then differentiate and u should get the derivative of t ie. the derivative of your expression.

6. for the second derivative differetiate again as you already might be knowing!

7. The second derivate

$\displaystyle f'(x)=6e^{6e^x+x}$
$\displaystyle f''(x)=6e^{6e^x+x} \cdot (6e^x+x)'$
$\displaystyle f''(x)=6e^{6e^x+x} \cdot (6e^x+1)$

8. Originally Posted by BugzLooney
Yes that is the question indeed.
If $\displaystyle g$ is a differentiable function and $\displaystyle f(x)=e^{g(x)}$, then $\displaystyle f$ is differentiable and:
$\displaystyle f'(x)=e^{g(x)} g'(x)$.
Just apply this on your problem.