[Math] Partial Derivative using limit definition

Question

Find the partial derivative in respect to $y$ of $f(x,y,z)=x^y$ using the limit definition.

My attempt:

$$\lim\limits_{h \to 0} \frac{f(x,y+h,z)-f(x,y,z)}{h}$$
$$\lim\limits_{h \to 0} \frac{x^{y+h}-x^y}{h}$$
$$\lim\limits_{h \to 0} \frac{x^yx^h-x^y}{h}$$

Now I am stuck because I don't know how to apply my log rules to this.

Answer 1

A few more steps:

$$\begin{align}\lim_{h\to0}\frac{x^yx^h-x^y}h&=x^y\lim_{h\to0}\frac{x^h-1}h\\&=x^y\lim_{h\to0}\frac{e^{h\ln(x)}-1}h\end{align}$$

Let $h\ln(x)=u$,

$$=x^y\lim_{u\to0}\frac{e^u-1}{\frac u{\ln(x)}}=x^y\ln(x)\lim_{u\to0}\frac{e^u-1}u$$

$$f_y=x^y\ln(x)$$