Differentiate products with the product rule

• Interpret the two product-rule terms as separate first-order changes.
• Differentiate a product without prematurely expanding it.
• Verify a product-rule derivative by comparing it with an expanded form.

Build the idea

Worked example

1. Step 1

   Let f equal x squared and g equal x plus three. Their derivatives are two x and one, respectively.

   $$f'=2x,\quad g'=1$$

2. Step 2

   Pair the derivative of the first factor with the original second factor, then add the original first factor times the derivative of the second.

   $$2x(x+3)+x^2(1)$$

3. Step 3

   Distribute and combine like terms. The derivative becomes three x squared plus six x.

   $$3x^2+6x$$

• Writing f prime times g prime as the derivative of the product.

  The rectangle changes through two strips, so retain one original factor in each term and add the contributions.

• Differentiating only one factor and leaving out the second term.

  Either factor can change with the input. The complete first-order change must account for both possibilities.

Differentiate the quantity x minus one times x cubed using the product rule and verify by expansion.

This reviewed reading order covers the lesson explanation, equations, example, and checks without claiming that a video exists.

1. Part 1

   Lesson overview

   When two changing factors are multiplied, the product changes through one factor at a time to first order. The product rule adds those two contributions. A rectangular area with side lengths f and g changes when either side changes. One thin strip has approximate area g times the change in f, and the other has area f times the change in g. Interpret the two product-rule terms as separate first-order changes.

2. Part 2

   Visual model and equations

   The tiny corner where both sides change is second order and disappears in the derivative limit. Dividing the remaining first-order change by the input change produces two added terms. Start with an f-by-g rectangle. Increasing both side lengths adds a vertical strip, a horizontal strip, and a tiny corner; the two strips supply the derivative terms. The derivative of f times g equals f prime times g plus f times g prime. The change in f g is approximately g times the change in f plus f times the change in g.

   • The derivative of f times g equals f prime times g plus f times g prime.
   • The change in f g is approximately g times the change in f plus f times the change in g.
3. Part 3

   Worked example

   Differentiate the product x squared times the quantity x plus three using the product rule, then verify by expansion. Name the two factors, differentiate each separately, apply both product-rule terms, and simplify before comparing with the expanded polynomial. Let f equal x squared and g equal x plus three. Their derivatives are two x and one, respectively. Pair the derivative of the first factor with the original second factor, then add the original first factor times the derivative of the second. Distribute and combine like terms. The derivative becomes three x squared plus six x. The derivative is three x squared plus six x. Expanding the original product gives x cubed plus three x squared. The power rule gives three x squared plus six x, matching the product-rule result.

   • f prime equals two x and g prime equals one.
   • Two x times the quantity x plus three, plus x squared times one.
   • Three x squared plus six x.
4. Part 4

   Checks, practice, and scope

   This is why the derivative of a product is not the product of derivatives. Both original factors remain, each paired once with the derivative of the other. Writing f prime times g prime as the derivative of the product. The rectangle changes through two strips, so retain one original factor in each term and add the contributions. Differentiating only one factor and leaving out the second term. Either factor can change with the input. The complete first-order change must account for both possibilities. Differentiate the quantity x minus one times x cubed using the product rule and verify by expansion. Use derivatives one and three x squared, then simplify both product-rule terms. The derivative is four x cubed minus three x squared. The product rule adds the first-order changes caused by each factor. Each term keeps one original factor and differentiates the other. Expansion offers an independent verification when the product is polynomial. The geometric strip argument is intuitive rather than a complete limit proof and assumes differentiable factors. Products of three or more factors require repeated application or a generalized sum of factorwise contributions.

Scope and limitations

• The geometric strip argument is intuitive rather than a complete limit proof and assumes differentiable factors.
• Products of three or more factors require repeated application or a generalized sum of factorwise contributions.