# Finding partial derivative $f_x$ of $\ln(x^2y)$ using definition.

How do find partial derivative $f_x$ of $f(x,y)=\ln(x^2y)$ using definition? I know that the answer is $\frac2x$, but I can't see how to get there by using limit $$\lim_\limits{\Delta x\to 0}\frac{\ln((x+\Delta x)^2y)-\ln(x^2y)}{\Delta x}.$$

• You can start by using the properties of the $\ln$ function. First, $\ln((x+\Delta x)^2y)=\ln(x+\Delta x)^2+\ln y$. Do the same with $\ln (x^2y)$... – Bernard Massé Mar 12 '17 at 18:49

which is recognized as the derivative of $2\ln(x)$ for $\Delta x\rightarrow 0$.