91Ó°ÊÓ

Use Definition 12.30 as a model for finding a solution for the given question.

The Definition 12.30 in the text describes the differentiability for functions of three variables.

Using this definition as a model, let's solve the given question.

Let $$f(x)$$ be the function of $$n$$ variables.

And, let the function, $$y=f(x)$$ be defined on an open set containing the point $$x_{0}$$.

Also, let $$\bigtriangleup y=f(x_{0}+\bigtriangleup x)-f(x_{0})$$

We know that the function $$f$$ is said to be differentiable at $$x_{0}$$ if the partial derivatives f_{x_{i}}(x_{0}) exist for each 1\leq i\leq n.

Hence, $$\bigtriangleup y= \langle f_{x_{1}}(x_{0}),f_{x_{2}}(x_{0}),...,f_{x_{n}}(x_{0})\rangle\cdot \bigtriangleup x+\epsilon \cdot \bigtriangleup x$$, where $$\epsilon \rightarrow 0$$ as $$\bigtriangleup x\rightarrow 0$$.