Key takeaway: The product rule gives the slope formula for f times g: take f-prime times g, then add f times g-prime, so that each factor is differentiated once while the other waits.
Grade 12 – AP Calculus ABAP Calculus AB Unit 2Calculus
Definition of Product Rule
The product rule differentiates an expression made by multiplying two functions of \(x\). Each factor changes as \(x\) changes, and both changes affect the product, so the derivative has two parts.
Think of a rectangle whose width is \(f\) and height is \(g\). If the width grows a little, the area gains a strip; if the height grows a little, it gains another strip. Adding those two gains gives the rate of change of the area.
Formula and notation
If \(y=f(x)\,g(x)\), then
\[ y^{\prime}=f^{\prime}(x)\,g(x)+f(x)\,g^{\prime}(x). \]
Here \(f\) and \(g\) are differentiable functions. To remember it, work on one factor, copy the other, then swap roles and add.
Key facts
- The answer always has two terms, one from differentiating each factor.
- Never multiply the two separate derivatives; that shortcut gives wrong slopes.
- When a factor is a constant, the product rule collapses to the constant multiple rule.
- Expanding first is a valid check for polynomials, but the product rule is needed for products like x times sin x.
Product Rule example
Differentiate \(y=(3x^2+1)(x^3-4x)\).
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Let \(f=3x^2+1\) and \(g=x^3-4x\). Then \(f^{\prime}=6x\) and \(g^{\prime}=3x^2-4\).
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Apply the rule: \(y^{\prime}=6x(x^3-4x)+(3x^2+1)(3x^2-4)\).
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Expand: \(6x^4-24x^2+9x^4-12x^2+3x^2-4\), which simplifies to \(15x^4-33x^2-4\).
The derivative is \(y^{\prime}=\)\(15x^4-33x^2-4\), which matches expanding \(3x^5-11x^3-4x\) first.
Common mistake
Watch out
For the same function a student writes \(y^{\prime}=6x\cdot(3x^2-4)=18x^3-24x\), multiplying the two derivatives.
Why it is wrong
The product rule is a sum of two terms, and each term keeps one original factor. Multiplying only \(f^{\prime}\) by \(g^{\prime}\) ignores how \(f\) and \(g\) themselves contribute. Test the claim on \(x\cdot x=x^2\): the derivative is \(2x\), but \(1\cdot 1=1\).
Related terms and lessons
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