Product Rule
In short: Product rule finds the derivative of two functions multiplied together: differentiate the first and keep the second, then keep the first and differentiate the second, and add the results.
AP Calculus ABCalculusAP Calculus AB Unit 2
What is Product Rule?
The product rule exists because, when two changing quantities are multiplied, the result changes for two reasons at once: the first one moves, and the second one moves. The rule adds both effects.
For differentiable functions \(u\) and \(v\):
\[\frac{d}{dx}\left[u\,v\right] = u^{\prime}v + u\,v^{\prime}\]
Picture a rectangle with sides \(u\) and \(v\) that are both growing. The extra area comes from a thin strip along each side, and the tiny corner square is negligible.
How to spot it
- The function is two expressions in x multiplied, such as (2x^3 – x)(x^2 + 5) or x times sin x.
- Expanding the product first would be long or impossible, for example when a trig function is one factor.
- A problem gives the values of f, g, f’ and g’ at one input and asks about the derivative of f times g.
- A rate depends on a product, such as revenue equals price times quantity with both changing.
The method
- Name the two factors: \(u\) is the first expression and \(v\) is the second.
- Differentiate each one separately to get \(u^{\prime}\) and \(v^{\prime}\).
- Assemble \(u^{\prime}v + u\,v^{\prime}\), writing the original factors untouched.
- Expand and combine like terms only at the end, once the product rule has done its work.
Product Rule: solved example
Problem
Differentiate \(y = (2x^3 – x)(x^2 + 5)\).
Solution
Let \(u = 2x^3 – x\), so \(u^{\prime} = 6x^2 – 1\). Let \(v = x^2 + 5\), so \(v^{\prime} = 2x\).
Then \(y^{\prime} = (6x^2 – 1)(x^2 + 5) + (2x^3 – x)(2x) = 6x^4 + 29x^2 – 5 + 4x^4 – 2x^2\).
Combine: \(y^{\prime} = 10x^4 + 27x^2 – 5\). As a check, expanding first gives \(2x^5 + 9x^3 – 5x\), which has the same derivative.
Try it
Question 1
Find the slope of \(y = x^2(4x – 1)\) at \(x = 2\).
Show the answer
\(y^{\prime} = 2x(4x – 1) + x^2(4)\). At \(x = 2\): \(4(7) + 4(4) = 28 + 16 = 44\).
Question 2
Differentiate \(f(x) = x \sin x\).
Show the answer
With \(u = x\) and \(v = \sin x\): \(f^{\prime}(x) = 1 \cdot \sin x + x \cos x = \sin x + x \cos x\).
Product Rule vs. Chain Rule
The product rule is meant for two expressions multiplied side by side, like \(x^2 \cdot \sin x\). The chain rule is for one expression placed inside another, like \(\sin(x^2)\). Ask yourself whether you see a times sign between two functions, or a function whose input is another function.
Keep going
Lessons
Worksheets and quizzes
- Product Rule, Quotient Rule and Trig Derivatives – Worksheet
- Product Rule, Quotient Rule and Trig Derivatives – Quiz
Related words
Whole dictionary: A–Z · AP Calculus AB · Calculus