Example 6:
In order to take the derivative of , you have to use the Chain Rule.
THE CHAIN RULE
The most important rule in this chapter (and sometimes the most difficult one) is called the Chain Rule.
It’s used when you’re given composite functions—that is, a function inside of another function. You’ll
always see one of these on the AP Exam, so it’s important to know the Chain Rule cold.
A composite function is usually written as: f(g(x)).
For example: If f(x) = and g(x) = , then f(g(x))= .
We could also find g(f(x)) = .
When finding the derivative of a composite function, we take the derivative of the “outside” function, with
the inside function g considered as the variable, leaving the “inside” function alone. Then, we multiply
this by the derivative of the “inside” function, with respect to its variable x.
Here is another way to write the Chain Rule.
The Chain Rule: If y = f(g(x)), then y′ = This rule is tricky, so here are several examples. The last couple incorporate the Product Rule and the
Quotient Rule.
Example 7: If y = (5x^3 + 3x)^5 , then = 5(5x^3 + 3x)^4 (15x^2 + 3)
We just dealt with the derivative of something to the fifth power, like this: