Partial Derivative $x^y$? in terms of x

Question

Hi i have the answer but don't understand so please explain your answer... it should be $-yx^{(y-1)}$

Answer 1

The definition of the partial derivative of $x^y$ with respect to $x$ is

$$\frac{\partial }{\partial x} x^y = \lim_{h\to 0} \frac{(x+h)^y - x^y}{h}.$$

In this way, we may simply treat $y$ as a constant when differentiating. So, $$\frac{\partial }{\partial x} x^y = yx^{y-1}$$ for the same reason that $$\frac{d }{d x} x^n = nx^{n-1}.$$

Answer 2

It should be $\frac{d}{{dx}}\left( {{x^y}} \right) = y{x^{y - 1}}$