Question

Would the derivative of y = (x^e)(e^x) be y' = (x^e)(e^x) + (e^x)(ex) ?

Answer 1

\[y=x^ee^x\]
By the product rule, the derivative is
\[y'=\frac{d}{dx}\left[x^e\right]e^x+x^e\frac{d}{dx}\left[e^x\right]\\
y'=ex^{e-1}e^x+x^ee^x\\
y'=x^{e-1}e^{x+1}+\color{red}{x^ee^x}\]
You have the red term correct, but the first one is wrong. Do you see why? Simply application of power rule.

Answer 2

I'm not sure I understand the third part. Where does the e from \[ez ^{e-1}\] go? And I don't understand where the +1 came from on e^x either. ><;

Answer 3

I used some exponent properties to rewrite:
\[a\times a^b=a^1\times a^b=a^{b+1}\]
So,
\[\large e\times x^{e-1}\times e^x=\color{blue}{x^{e-1}\times e^{x+1}}\]
(blue is what I have in the last line)

Answer 4

Ohhhh, that took me a while. lol. Thank you very much for your time and patience. :D