The Chain Rule

Autor:Ku, Yin Bon (Albert)

Chain rule There are several versions of chain rule that can help us computing the derivatives/partial derivatives of composition of functions that involve several variables. Composition of a vector-valued function and a multivariable function Let

and

. Then we consider their composition:

Heuristically, we have

, we have the following chain rule:

(Note: After computing

and

, you still need to substitute

and

in order to get the right hand side in terms of

only.) The three-variable version of the chain rule is similar. Let

and

. Then we have

Example: Let

and

. Find

. Answer: First of all,

and

. We also have

and

. Then by chain rule, we have

Alternatively, you can compute

with using chain rule by first writing down

directly:

Then we have

Exercise: Let and . Find .

The applet below visualizes the composition of

and

. The point Q along the curve on the graph of

has coordinates

. And

measures the rate of change of

as the curve on the xy-plane, which is parametrized by

, is traversed.

Composition of functions of two variables Suppose

and

. Then we consider their composition:

The following is the chain rule for

The three-variable version of the above chain rule is similar. Let

and

. Then we have

Example: Let

, and

. Find

. Answer:

Example: Let

be a constant. Suppose

and

. Find

and

. Answer:

Exercise: Let and . Find and .

Implicit differentiation We can use chain rule to derive the general formula for

when

and

are related by the equation

, where

is a differentiable function of two variables. We let

and

i.e.

is implicitly defined as a function of

. Then we have

Since

and

. Hence we have

Example: Suppose

. Find

. Answer: Let

and

. By the above formula, we have

The Chain Rule

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