Tutoring calculus, your students will need to know this.  The tutor introduces it.

Once a calculus student learns the chain rule, they need to use it virtually all the time. It’s like learning to read when you’re a kid:  once you know how, you’re expected to do it forever.

The chain rule is a quintessential technique of calculus:  it allows you to easily find derivatives you could barely hope to accomplish any other way.  Its fundamental meaning is as follows:

Let’s imagine you have two functions, f(x) and g(x). They have been composed into the function f(g(x)).

The derivative of f(g(x)) is the derivative of the outer one, still at g(x), multiplied by the derivative of the inner one. Behold:

d/dx f(g(x))=f'(g(x))g'(x)

Example 1:

Find the derivative of (3x^2-7)^11.

Without the chain rule, this derivative would be hopeless to attempt. With the chain rule, it’s straightforward.

Step 1: Realize which is the outer function versus which is the inner function.

In this case, the outer function is f(x)=(g(x))^11. The inner function is g(x)=3x^2-7

Step 2: Find the derivative of the outer function.

d/dx (g(x))^11 = 11(g(x))^10

Step 3: Find the derivative of the inner function.

The derivative of the inner function is easily found as well, using the power rule and the constant rules:

d/dx (3x^2-7)=6x

Step 4: Multiply the outer and inner derivatives to get the final answer:

d/dx (3x^2-7)^11 = (11(3x^2-7)^10)*6x

You may now want to simplify:

d/dx =66x(3x^2-7)^10

Example 2:

Find the derivative of f(x)=cos(x^7)

Solution:

Step 1: Obviously, the outer function is cos(inner function); the inner function is x^7.

Step 2: We find the derivative of the outer function:

(cos(inner function))’=-sin(inner function)=-sin(x^7)

Step 3: We find the derivative of the inner function:

(x^7)’=7x^6

Step 4: We multiply the outer and inner derivatives together:

-sin(x^7)*7x^6

You may now want to simplify:

-7x^6sin(x^7)

I’ll be talking more about the chain rule in future posts. HTH:)

Source:

Larson & Hostetler: Calculus, part one, third ed. Toronto: D.C. Heath and Company, 1989.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.