Partial Derivative of $x^y$

Question

$$\frac{\partial}{\partial x}x^y$$

$x^y=e^{ln(x)^y}=e^{y*ln(x)}$

$$\frac{\partial}{\partial x}e^{y\cdot ln(x)}=e^{y\cdot ln(x)}\cdot\frac{y}{x}=x^y\cdot\frac{y}{x}=x^{y-1}\cdot y$$

But the answer is different , where did I get it wrong?

Answer 1

You have the correct answer. WolframAlpha thinks $y$ is a function of $x$, which is why it gives the answer it does. Try using another variable like $t$ instead of $y$ and you'll see that WolframAlpha gives the right answer.

As a side note, I noticed that the query you typed into WolframAlpha was "derive x^y with respect to x". The process of finding a derivative is differentiation, not derivation, so it would be more correct to ask "WolframAlpha to "differentiate x^y with respect to x". The two terms are (incorrectly) used interchangeably so often that WolframAlpha knows you mean "differentiate" when you say "derive".

Answer 2

You didn't get it wrong: Wolfram Alpha just doesn't know that $y$ doesn't depend on $x$.

If it did, then the term that comes from using the product rule would be $$x \log(x) y'=0$$