The problem I have are from the first partial derivatives of $f(x,y) = x^y$
What is its $f_x(x,y)$ and $f_y(x,y)$?
I need to find the critical points of $f(x,y) = x^y + 4xy – 2y^2 + 5$, but the $x^y$ is making me confused.
The answer I get when I try to find its partial derivatives are
$f_x(x,y) = yx^{y-1}+4y$
$f_y(x,y) = x^y\ln x+4x-4y$
I am stuck in this step and I am not sure if my partial derivatives are right.
Best Answer
if our function is:
$$ f(x,y) = x^y$$
Then,
$$ \frac{ \partial f}{\partial x} = y x^{y-1}$$
And,
$$ \frac{ \partial f}{ \partial y} = \frac{ \partial x^y}{\partial y}= \frac{ \partial}{\partial y} e^{ y \ln x} = e^{ y \ln x} \frac{\partial (y \ln x)}{\partial y} $$