Math · Calculus · Concept
Why the product rule has two terms
A changing rectangle gains two edge strips and one smaller corner.
Account for both changing factors
For positive side lengths, a rectangle makes the product visible. Change its sides by Δf and Δg. The new area contains the old rectangle, two strips, and a corner. The same expansion is an algebraic identity for signed quantities.
Divide by the input change
When f and g are differentiable at x, their increments tend to zero. Divide the identity by h. The corner contribution is (Δf/h)Δg, which tends to f′(x)·0; each strip leaves one finite derivative term.
Keep the unchanged partner
Each derivative measures one factor changing while the other supplies its current scale. Differentiability at the point is the hypothesis; merely multiplying f′ by g′ accounts for neither strip.
Common mistakes
- The product derivative is not f′g′.
- Do not omit the small corner before taking the limit; it is part of the exact finite change.
Work through an example
Find the derivative in product form and check by expansion.
Differentiate x²(x+1) →Sources and scope
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