Welcome back to AP Calculus with Fiveable! This topic focuses on taking the derivative of a product. Weโve worked through derivatives at a point, sum and difference rules, and trigonometric derivatives, so let's keep building our derivative skills. ๐
๐ย Product Rule Definition
To find the derivative of a product of functions, we need to multiply the first function by the derivative of the second and add it to the second function multiplied by the derivative of the first.
If we incorrectly attempt to calculate the derivative of f(x), it would say
fโฒ(x)=cos(x)(2x+2)
However, sin(x)(2x+2)+(x2+2x)cos(x)๎ =cos(x)(2x+2).
This can be seen in the following graphs. fโฒ(x) represents the correct derivative of f(x) because the critical points and positive and negative values match the original functions.
Graph of f(x)
Graph of f(x) created with Desmos
Graph of fโฒ(x)
Graph of fโฒ(x) created with Desmos
Incorrect Graph of fโฒ(x)
Incorrect Graph of fโฒ(x); Graph created with Desmos
๐งฎย Product Rule: Practice Problems
Letโs work on a few questions and make sure we have the concept down!
Product Rule: Example 1
Find yโฒ for y=(3x2โ4x)(2xโ1)
with and without the Product Rule.
Solving Example 1 Without Product Rule
To find yโฒ without the product rule, we have to first expand the function.
Unless specified, you do not have to simplify for the AP Calculus Exam, so this answer is perfectly acceptable! โ
Product Rule: Example 2
Find fโฒ(x) if f(x)=sin(x)(3x2โ2x+5).
Let's use the product rule to find the derivative of f(x). Donโt forget, the derivative of sin(x) is cos(x)! Brush up on your trig derivatives with this Fiveable guide: Derivatives of cos x, sinx, e^x, and ln x.
Find yโฒ if y=[ex](https://www.fiveableKeyTerm:ex)sin(x)
Remember that the derivative of ex is still ex! Now we can use the product rule.
yโฒ=exโdxdโ(sin(x))+sin(x)โdxdโ(ex)
Therefore,
yโฒ=excos(x)+exsin(x)
๐ย Closing
Great work! ๐ย The product rule is a key foundational topic for AP Calculus. You can anticipate encountering questions involving the product rule on the exam, both in multiple-choice and as part of a free response.