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We begin with functions of one variable. We say a function $f(x)$ is
You may wonder why we didn't just say `a function is differentiable at $(a,b)$ if its partial derivatives exist at $(a,b)$?'
So for a function to be differentiable, we need more than just the existence of partial derivatives. We need continuity of partial derivatives:
The composition of differentiable functions is differentiable. In particular, if $x(t)$ and $y(t)$ are differentiable functions of a parameter $t$, and if $f(x,y)$ is a differentiable function of $x$ and $y$, then $f(x(t),y(t))$ is a differentiable function of $t$. In the next slide, we'll compute its derivative. |