We investigate the chain rule for functions of several variables.
It is good to understand what the situation of , describes. We know that describes a surface; we also recognize that describes a curve in the -plane. Combining these together, we are describing a curve that lies on the surface described by . The parametric equations for this curve are , and . Consider:
Now try your hand at the chain rule.
The previous example can make us wonder: if we substituted for and at the end to show that is really just a function of , why not substitute before differentiating, showing clearly that is a function of ?
That is, Applying the chain and product rules, we have which matches the result from the example.
This may now make one wonder “What’s the point? If we could already find the derivative, why learn another way of finding it?” In some cases, applying this rule makes differentiation simpler, but this is hardly the power of the chain rule. Rather, the chain rule is extremely powerful when we do not know what , and/or are. It may be hard to believe, but often in “the real world” we know rate-of-change information (information about derivatives) without explicitly knowing the underlying functions. The chain rule allows us to combine several rates of change to find another rate of change.
The chain rule also tells us something about the meaning of the gradient. As we will see, the gradient vector is always orthogonal to level curves and surfaces.
Note that the last explanation works in any dimension. The up-shot?
Gradient vectors are orthogonal to level sets.
This is a key concept concerning the gradient.
We can also use our new chain rule to revisit problems from our previous studies of calculus. Our new tools allow for simpler solutions to these problems.
Recall the following form of the Fundamental Theorem of Calculus:
We’ve used implicit differentiation to compute when is given as an implicit function of . Now we’ll revisit this with the chain rule and give a new, simpler, method of finding .
For instance, consider the implicit function . We learned to use the following steps to find :
Instead of using this method, consider . The implicit function above describes the level curve . Considering and as functions of , the chain rule states that Since is constant (in our example, ), . We also know . Write with me,
Note how our solution for above is just the partial derivative of , with respect to , divided by the partial derivative of with respect to . We state the above as a theorem.
Try your hand at this.