How can I show that $f$ is differentiable at $\vec 0$, where
$$
f(x,y)=x^2\log(x^4+y^2),
$$
and $f(0,0)=0$. I’ve already shown that $f$ is continuous at $\vec 0$. I started off by calculation the first partial derivative:
$$
D_1f(\vec 0)=\lim_{t\to 0}t\log t^4,
$$
however, I don’t know how to calculate this limit even. I looked at the plot, and it seems that $D_f(\vec 0)=D_2f(\vec 0)=0$, so apparently $t$ goes faster to zero then $\log t^4$ goes to minus infinity. How can I show this? Can I use Taylor? Should I evaluate then at $x=1$? This would yield: