Chain Rule application:

A snowball has volume where r is the radius. The snowball is melting so that at the instant that the radius is 4 cm. the radius is decreasing at the rate of .25 cm/min.

What is the rate of change of the volume at this instant?

By the chain rule, So when r=4 and we have

cubic cm/min

Another example:

Given and x is a function of t for which x(1)=2 and x’(1)=0.3,

find dy/dt when t=1. (Note: x is 2 if t=1 and dx/dt is 0.3 if t=1)

Plug in x=2 and dx/dt =0.3 to get dy/dt at t=1 is .

Reversing the Chain Rule/ Substitution in antidifferentiation

When you see a composite you differentiate it using the chain rule.

This means

To go backwards, you have the derivative and want the antiderivative.

Substitute u(x)=the inside of the composite. Here u=

We will change the integrand (the function inside the integral) to a function of u and replace u’(x)dx with du.

Here since

Then

Ex.

We do not have the factor of 3 but that can be fixed. Just multiply and divide by 3.

You can always check your answer by differentiating the result and you should get the integrand back. We were lucky that we just happened to have a constant multiple of du.

Substitution is only one method of finding antiderivatives and does not always work. It is the only method besides reversing the power rule and doing algebra that we will learn.

One more example. Sometimes we use substitution just to rearrange the product so we can multiply more easily.

Ex. If the sum or difference were outside the square root and only x were on the inside, we could multiply. Ex. Then we can just reverse the power rule.

So we substitute u=x+1. x=u-1 and du=dx Now we have