partial derivatives $x^y$

Question

Question: Find the first-order partial derivatives and second-order partial derivatives for: $f(x,y) = x^y$

I understand $\frac{\partial f}{\partial x} = yx^{y-1}$

Can someone explain why $ \frac{\partial f}{\partial y} = x^{y}\log{x}$

Answer 1

For a partial derivative you hold one variable constant while you differentiate with respect to another. So $$ \frac{\partial f}{\partial y}=x^y\ln x $$ is equivalent to $$ \frac{\text d}{\text dx}a^x=a^x\ln a $$

Answer 2

HINT

Let

$$f(x,y) = x^y=e^{y\log x}$$

and use

$$[e^{g(y)}]'=e^{g(y)}\cdot g'(y)$$