Calc Question, Will Medal! Find y' if y=(x^e)(e^x)

I understand the first portion of the problem (product rule) but having trouble with other parts.

Question

Calc Question, Will Medal! Find y' if y=(x^e)(e^x)

I understand the first portion of the problem (product rule) but having trouble with other parts.

anonymous · Answer

$$y=x^{e}e^{x}$$

zepdrix · Answer

Oo that's a cool problem :)

anonymous · Answer

Hahaha

zepdrix · Answer

Start with product rule, ya?$$\large\rm y'=(x^e)'e^x+x^e(e^x)'$$

zepdrix · Answer

Derivative of e^x?

anonymous · Answer

The derivative of e^x would be e^x

zepdrix · Answer

$$\large\rm y'=(x^e)'e^x+x^e(e^x)$$Ok cool.

anonymous · Answer

I am franticly looking through my notes....

anonymous · Answer

Sorry :(

zepdrix · Answer

It's simpler than you think :)
$\large\rm \frac{d}{dx}x^n=nx^{n-1}$

anonymous · Answer

Ohhhhh!!! So that means I can simply do:

$$y' = (ex ^{e-1})(e ^{x}) + (e ^{x})(x ^{e})$$

zepdrix · Answer

Yes, good job.
From that point, you could do some factoring,
not completely necessary though.

anonymous · Answer

Ok, would I next have to get the natural log of parts of the equation?

anonymous · Answer

I take that back... that is pretty much the answer isn't it?

zepdrix · Answer

Mmmmm, no.
No logs required.

But as I said, you could factor if you like,$$\large\rm y'=ex^{e-1}e^x+x^ee^x$$$$\large\rm y'=x^{e-1}e^x(e+x)$$

zepdrix · Answer

Yes, you pretty much have your answer :))

anonymous · Answer

Haha thank you so much for you time. I appreciate it!

zepdrix · Answer

np