2008-22.md

course	course_year	question_number	tags	title	year
Analysis II	IB	22	IB 2008 Analysis II	$3 . \mathrm{I} . 3 \mathrm{~F} \quad$	2008

Explain what it means for a function $f(x, y)$ of two variables to be differentiable at a point $\left(x_{0}, y_{0}\right)$. If $f$ is differentiable at $\left(x_{0}, y_{0}\right)$, show that for any $\alpha$ the function $g_{\alpha}$ defined by

$$g_{\alpha}(t)=f\left(x_{0}+t \cos \alpha, y_{0}+t \sin \alpha\right)$$

is differentiable at $t=0$, and find its derivative in terms of the partial derivatives of $f$ at $\left(x_{0}, y_{0}\right)$.

Consider the function $f$ defined by

Is $f$ differentiable at $(0,0)$ ? Justify your answer.