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Product & Quotient Rules

Overview

Multiplying changes.

What is it?

Formulas for finding the derivative when two functions are multiplied or divided.

History

Leibniz struggled with this, initially thinking the derivative of a product was just the product of the derivatives (it's not!).

Key Idea

Product: Left D-Right + Right D-Left. Quotient: Lo D-Hi minus Hi D-Lo, over Lo Lo.

Practice This Topic

Concept Guide

Plain English: When two dynamic things interact (like Price changing AND Sales Volume changing), you can't just look at them separately. You have to account for how the first affects the second and vice versa.

Real-world example: Economics (Total Revenue). Revenue = Price * Quantity. Since both Price and Quantity change, you need the Product Rule to find the rate of revenue growth.

How to do it

  1. Identify your two functions, f and g.
  2. Find f' (derivative of f) and g' (derivative of g).
  3. For Product: f * g' + g * f'.
  4. For Quotient: (g * f' - f * g') / g^2.
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Common Pitfall

Quotient Rule order matter! Because it involves subtraction, you must do (Bottom * DerivTop) FIRST.

Word Problem
"A rectangle's length 'x' is growing at 2 m/s, and height 'y' is growing at 3 m/s. At the moment x=10 and y=5, how fast is the Area changing?"
Reasoning: Area = x * y. Use Product Rule: x(y') + y(x'). Plug in values: 10(3) + 5(2) = 30 + 10 = 40 m²/s.

Practice Examples

Product Rule
d(uv) = u'v + uv'
Mix and match derivatives.