Math 10C - Spring 2001
Chain rule

The chain rule can be visualized by drawing a hiking trail on a contour map.

Suppose you hike on a mountain shaped like a hemisphere , as shown above in 3D. The contour map representing it is at the bottom of the same figure.

You hike on a trail given by a parametric curve , . This is drawn in black on the contour map, and then lifted up to the surface. This turns z into a function of t:

You can compute dz/dt (the rate of change of elevation) in two ways.

• By the chain rule,

  

• Directly differentiating gives the same result:

  

This is a contrived example where it is possible to do it two ways and compare them, and the old way is easier. With other problems, the chain rule might be easier or might be the only way to solve them.

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