We introduce partial derivatives.

Given a function , it is often useful to differentiate with respect to a single variable and hold the other variables as constants. As a concrete example, let Fixing , allows us to focus our attention to all points on the surface where the -value is ,
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We can now focus our attention on the curve
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and differentiate this curve purely with respect to .
Let . Compute:
Compute

Let . Compute:
Compute:

We have shown how to compute a partial derivative, but it may still not be clear what a partial derivative means. Given , measures the rate at which changes as only varies: is held constant.

Imagine standing in a rolling meadow, then beginning to walk due east. Depending on your location, you might walk up, sharply down, or perhaps not change elevation at all. This is similar to measuring : you are moving only east (in the ‘‘’’-direction) and not north/south at all. Going back to your original location, imagine now walking due north (in the ‘‘’’-direction). Perhaps walking due north does not change your elevation at all. This is analogous to : does not change with respect to . We can see that and do not have to be the same, or even similar, as it is easy to imagine circumstances where walking east means you walk downhill, though walking north makes you walk uphill. The next example helps us visualize this.

Whenever we do a computation in mathematics, we should ask ourselves, ‘‘What does this mean?’’

What is the meaning of

First note that . If , this means if one ‘‘stands’’ on the surface at the point and moves

parallelorthogonal to the -axis (so only the -value changes, not the -value), then the instantaneous rate of change in is . Increasing the -value will increasedecrease the -value; decreasing the -value will increasedecrease the -value.
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If , this means if one ‘‘stands’’ on the surface at the point and moves

parallelorthogonal to the -axis (so only the -value changes, not the -value), then the instantaneous rate of change in is . Increasing the -value will increasedecrease the -value; decreasing the -value will increasedecrease the -value.
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Finally, since the magnitude of is greater than the magnitude of at , the surface is ‘‘steeper’’ in the -direction than in the -direction.

Estimating partial derivatives

Functions of several variables, especially ones that map can be described by a table of values or level curves. In either case we can estimate partial derivatives by looking at Let’s do an example to make this more clear.

Let be a differentiable function described by the following table of values:
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Estimate
Work as we did in the example above, finding two estimates and taking their averages.

We can also estimate partial derivatives by looking at level curves.

Let be described by the level curves below:
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The height of the level curve is marked on the curve, and we are given a point . Estimate
Work as we did in the example above, finding two estimates and taking their averages.

Combining partial derivatives

While a function only has one second derivative. However, functions have second partial derivatives and functions have second partial derivatives! Don’t run off yet, things get better.

The notation of second partial derivatives gives some insight into the notation of the second derivative of a function of a single variable. If , then . The ‘‘’’ portion means ‘‘take the derivative of twice,’’ while ‘‘’’ means ‘‘with respect to both times.’’ When we only know of functions of a single variable, this latter phrase seems silly: there is only one variable to take the derivative with respect to. Now that we understand functions of multiple variables, we see the importance of specifying which variables we are referring to.

Consider: Find six first and second partial derivatives.

Notice how above . The next theorem states that it is not a coincidence.

Finding and independently and comparing the results provides a convenient way of checking our work.

Differentiability

In the past you may have learned

Given a function and a number in the domain of , if one can ‘‘zoom in’’ on the graph at sufficiently so that it appears to be a straight line, then the function is differentiable, and that line is the tangent line to at the point .

We illustrate this informal definition with the following diagram:

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When working with functions of several variables, intuitively, a function is differentiable when one can ‘‘zoom-in’’ and the surface determined by looks like a plane.

Given a function and a vector in the domain of , if one can ‘‘zoom in’’ on the graph at sufficiently so that it appears to be a plane, then the function is differentiable, and that plane is the tangent plane to at the point .

The following theorem states that differentiable functions are continuous, followed by another theorem that provides a more tangible way of determining whether a great number of function are differentiable or not.

The theorems assure us that essentially all functions that we see in the course of our studies here are differentiable (and hence continuous) on their natural domains. There is a small caveat to the Theorem above: it is possible for a function to be differentiable yet some partials may not be continuous. Such strange behavior of functions is a source of delight for many mathematicians.

Finding tangent planes

In your earlier calculus courses, you often found tangent lines to curves. To do this, you were given a function , a point , and then you produced as your tangent line. For functions , and a point we find tangent planes. These are planes of the form:

Find a tangent plane to at .