This is a fairly short section and is here so we can
acknowledge that the two main interpretations of derivatives of functions of a
single variable still hold for partial derivatives, with small modifications of
course to account of the fact that we now have more than one variable.
The first interpretation we’ve already seen and is the more
important of the two. As with functions
of single variables partial derivatives represent the rates of change of the
functions as the variables change. As we
saw in the previous section, represents the rate of change of the function as we change x and hold y fixed while represents the rate of change of as we change y and hold x fixed.
Example 1 Determine
if is increasing or decreasing at ,
we allow x to vary and hold y fixed.
we allow y to vary and hold x fixed.
(a) If we allow x to vary and hold y fixed.
In this case we will first need and its value at the point.
So, the partial derivative with respect to x is positive and so if we hold y fixed the function is increasing at as we vary x.
(b) If we allow y to vary and hold x fixed.
For this part we will need and its value at the point.
Here the partial derivative with respect to y is negative and so the function is
decreasing at as we vary y and hold x fixed.
Note that it is completely possible for a function to be
increasing for a fixed y and
decreasing for a fixed x at a point
as this example has shown. To see a nice
example of this take a look at the following graph.
This is a graph of a hyperbolic
paraboloid and at the origin we can see that if we move in along the y-axis the graph is increasing and if we
move along the x-axis the graph is
decreasing. So it is completely possible
to have a graph both increasing and decreasing at a point depending upon the
direction that we move. We should never
expect that the function will behave in exactly the same way at a point as each
The next interpretation was one of the standard
interpretations in a Calculus I class.
We know from a Calculus I class that represents the slope of the tangent line to at . Well, and also represent the slopes of tangent
lines. The difference here is the
functions that they represent tangent lines to.
Partial derivatives are the slopes of traces.
The partial derivative is the slope of the trace of for the plane at the point . Likewise the partial derivative is the slope of the trace of for the plane at the point .
Finally, let’s briefly talk about getting the equations of
the tangent line. Recall that the equation of a line in 3-D space is given by a vector
equation. Also to get the equation we
need a point on the line and a vector that is parallel to the line.
The point is easy.
Since we know the x-y coordinates of the point all we need
to do is plug this into the equation to get the point. So, the point will be,
The parallel (or tangent) vector is also just as easy. We can write the equation of the surface as a
vector function as follows,
We know that if we
have a vector function of one variable we can get a tangent vector by
differentiating the vector function. The
same will hold true here. If we
differentiate with respect to x we
will get a tangent vector to traces for the plane (i.e.
for fixed y) and if we differentiate
with respect to y we will get a
tangent vector to traces for the plane (or fixed x).
So, here is the tangent vector for traces with fixed y.
We differentiated each component with respect to x.
Therefore the first component becomes a 1 and the second becomes a zero
because we are treating y as a
constant when we differentiate with respect to x. The third component is
just the partial derivative of the function with respect to x.
For traces with fixed x
the tangent vector is,
The equation for the tangent line to traces with fixed y is then,
and the tangent line to traces with fixed x is,