Geometric Interpretation of Partial Derivatives

The picture to the left is intended to show you the geometric interpretation of the partial derivative.

The wire frame represents a surface, the graph of a function z=f(x,y), and the blue dot represents a point (a,b,f(a,b)). The colored curves are "cross sections" -- the points on the surface where x=a (green) and y=b (blue). The initial value of b is zero, so when the applet first loads, the blue cross section lies along the x-axis.

Recall the meaning of the partial derivative; at a given point (a,b), the value of the partial with respect to x, i.e.

fx(a,b)
is the slope of the line tangent to the blue cross section. (Change in z over change in x.) In other words, it tells you how fast z changes with respect to changes in x.

The value of fy(a,b), of course, tells you the rate of change of z with respect to y. That's the slope of the line tangent to the green curve.

Both of the tangent lines are drawn in the picture, in red. Click and drag the blue dot to see how the partial derivatives change. There's a lot happening in the picture, so click and drag elsewhere to rotate it and convince yourself that the red lines are actually tangent to the cross sections. You might have to look at it from above to see that the red lines are in the planes x=a and y=b!

In the next picture, we'll change things to make it easier on our eyes.

The cross sections and tangent lines in the previous section were a little disorienting, so in this version of the example we've simplified things a bit. We've replaced each tangent line with a vector in the line. Specifically, we're using the vectors
(1,0,fx(a,b))
(0,1,fy(a,b))
When you place these vectors at the point (a,b), they're contained in the tangent lines precisely because of the definition of the partial derivatives; for example, fx(a,b) tells us the change in z over the change in x in the tangent line. Hence, if we change x by 1, leave y alone, and change z by fx(a,b), we're just moving along the tangent line.

You can move the blue dot around; convince yourself that the vectors are always tangent to the cross sections. Also see if you can tell where the partials are most positive and most negative.

In the next picture we'll show how you can use these vectors to find the tangent plane.

A tangent plane is really just a linear approximation to a function at a given point. The partial derivatives fx(a,b) and fy(a,b) tell us the slope of the tangent plane in the x and y directions.

Put differently, the two vectors we described above,

(1,0,fx(a,b))
(0,1,fy(a,b))
are both parallel to the tangent plane. (Often, by an abuse of language, we say they're "in" the tangent plane, but that's really only true if you put the tails of these vectors at (a,b) or some other point in the plane. Ask your teacher if you're confused about this point.)

The picture on the left includes these vectors along with the plane tangent to the surface at the blue point. Once again, you can click and drag the point to move it around. See how the vectors are always in the plane?

This is a useful fact if we're trying to find a parametric equation of a tangent plane: the equation is simply

p(s,t)=(a,b,f(a,b))+s*(1,0,fx(a,b))+t*(0,1,fy(a,b)),
where s and t both range over the real numbers.

This page is http://www.math.umn.edu/~rogness/multivar/partialderivs.shtml

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The views and opinions expressed in this page are strictly those of the page author. The contents of this page have not been reviewed or approved by the University of Minnesota.

rogness@math.umn.edu