Most of the time, there is an easier way to conclude that a derivative exists and to find it. It involves the notion of a C^{1} function.
Definition 12.5.1 When f : U → ℝ^{p} for U an open subset of ℝ^{n} and the vector valued functions
It turns out that for a C^{1} function, all you have to do is write the matrix described in Theorem 12.3.3 and this will be the derivative. There is no question of existence for the derivative for such functions. This is the importance of the next theorem.
Theorem 12.5.2 Suppose f : U → ℝ^{p} where U is an open set in ℝ^{n}. Suppose also that all partial derivatives of f exist on U and are continuous. Then f is differentiable at every point of U.
Proof: If you fix all the variables but one, you can apply the fundamental theorem of calculus as follows.
 (12.7) 
Here is why. Let h

and so, taking the limit as h → 0 yields

Therefore,

Now I will use this observation to prove the theorem. Let v =

Then with this convention,
Some explanation of the step to the last line is in order. The messy thing at the end is o
Here is an example to illustrate.
Example 12.5.3 Let f
From Theorem 12.5.2 this function is differentiable because all possible partial derivatives are continuous. Thus

In particular,

Here is another example.
Example 12.5.4 Let f
All possible partial derivatives are continuous, so the function is differentiable. The matrix for this derivative is therefore the following 3 × 3 matrix

Example 12.5.5 Suppose f
Taking the partial derivatives of f, f_{x} = y,f_{y} = x,f_{z} = 2z. These are all continuous. Therefore, the function has a derivative and f_{x}

Also, for
When a function is differentiable at x_{0}, it follows the function must be continuous there. This is the content of the following important lemma.
Lemma 12.5.6 Let f : U → ℝ^{q} where U is an open subset of ℝ^{p}. If f is differentiable at x, then f is continuous at x.
Proof: From the definition of what it means to be differentiable,

and so, for y close enough to x, there exists a constant C such that

which shows that f is continuous at x. ■
There have been quite a few terms defined. First there was the concept of continuity. Next the concept of partial or directional derivative. Next there was the concept of differentiability and the derivative being a linear transformation determined by a certain matrix. Finally, it was shown that if a function is C^{1}, then it has a derivative. To give a rough idea of the relationships of these topics, here is a picture.
You might ask whether there are examples of functions which are differentiable but not C^{1}. Of course there are. In fact, Example 12.3.8 is just such an example as explained earlier. Then you should verify that f^{′}
Example 12.5.7 Find an example of a function which is not differentiable at
Here is a simple example.

To see this works, note that f is defined everywhere and

so clearly f is continuous at

and so f_{x}

and so f_{y}

does not even exist, much less equals 0. To see this, let x = y and let x → 0.
+ class=”left” align=”middle”(v)12.6. THE CHAIN RULE