Question

logarithmic differention: y=x^e^x

Answer 1

\[\huge y=x^{e^x}\]
Is this what the problem looks like? :)

Answer 2

Yes.

Answer 3

So we are not allowed to apply the power rule, since we have a VARIABLE in the EXPONENT. So we need to find a way to remove it from the exponent location.

Logs give us some neat little tricks for dealing with that.
So for the first step, we'll take the natural log of both sides.
\[\huge \ln y=\ln (x^{e^x})\]

Answer 4

\[\large \log(a^b)=b*\log(a)\]
So here's a rule of exponents that we're going to want to apply.

Answer 5

See how that will help us on the right side? :D
Hmm so let's apply it and see how our equation looks.
\[\huge \ln y=\ln (x^{(e^x)})=(e^x) \ln (x)\]

Understand what happened there? it's a lil tricky I know. :o

Answer 6

Slowly understanding! This is making sense.