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Product & Quotient Rules

The Hook

Multiplying changes.

What is it?

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

Time Travel (History)

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

Cheat Code

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

Practice This Now!

The Field Guide

In 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.

In The Real World: 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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Booby Trap!

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

Real World Challenge
"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?"
The Logic: Area = x * y. Use Product Rule: x(y') + y(x'). Plug in values: 10(3) + 5(2) = 30 + 10 = 40 m²/s.

Training Drills

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