# Partial Derivative using limit definition

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.

• Well, if you know it's a "log rule" (which I interpret as: "I know I should remember it from a very specific moment of my tuition history, but I don't") perhaps you'd better look it up in the high school book, no? – user228113 Feb 11 '17 at 17:49
• Your comment does not help me in any way,shape,or form. Thanks – combo student Feb 11 '17 at 18:08

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 $h\ln(x)=u$,

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

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

• why can we factor out an $x^y$ outside the limit? – combo student Feb 11 '17 at 17:47
• @combostudent Because it is independent of $h$?? – Simply Beautiful Art Feb 11 '17 at 17:48
• another question, if $u=hln(x)$ why can you write $u \rightarrow 0$? – combo student Feb 11 '17 at 17:58
• @combostudent if $h\to0$, what can you conclude about $u\to?$, assume $x>0$. – Simply Beautiful Art Feb 11 '17 at 18:21