doesn’t matter. It does matter with other rules, though, so it
helps to use the same order each time.If we were to simplify this, we’d get the same answer as before. But here’s the best part: We’re not going
to simplify it. One of the great things about the AP Exam is that when it’s difficult to simplify an
expression, you almost never have to. Nonetheless, you’ll often need to simplify expressions when you’re
taking second derivatives, or when you use the derivative in some other equation. Practice simplifying
whenever possible.
Example 1: f(x) = (9x^2 + 4x)(x^3 − 5x^2 )
f′(x) = (9x^2 + 4x)(3x^2 − 10x) + (x^3 − 5x^2 )(18x + 4)Example 2: y = ( + )(x^5 − 11x^8 )
y′ = ( + )(5x^4 − 88x^7 ) + (x^5 − 11x^8 )Example 3:
THE QUOTIENT RULE
What happens when you have to take the derivative of a function that is the quotient of two other
functions? You guessed it: Use the Quotient Rule.
The Quotient Rule: If f(x) = , then f′(x) = In this rule, as opposed to the Product Rule, the order in which you take the derivatives is very important,
because you’re subtracting instead of adding. It’s always the bottom function times the derivative of the
top minus the top function times the derivative of the bottom. Then divide the whole thing by the bottom
function squared. A good way to remember this is to say the following: