VG3
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Quiz
What is the product rule?
The product rule is used when differentiating a product of two functions.
When two functions are multiplied together, we cannot simply differentiate each separately. We must use the product rule.
[u · v]' = u' · v + u · v'
Differentiate the first and keep the second, plus keep the first and differentiate the second.
Memory tip: "Derivative-first times second, plus first times derivative-second"
Examples
Example 1: Differentiate h(x) = x² · sin(x)
u = x² → u' = 2x
v = sin(x) → v' = cos(x)
h'(x) = 2x · sin(x) + x² · cos(x)
v = sin(x) → v' = cos(x)
h'(x) = 2x · sin(x) + x² · cos(x)
Example 2: Differentiate h(x) = eˣ · ln(x)
u = eˣ → u' = eˣ
v = ln(x) → v' = 1/x
h'(x) = eˣ · ln(x) + eˣ · (1/x) = eˣ(ln(x) + 1/x)
v = ln(x) → v' = 1/x
h'(x) = eˣ · ln(x) + eˣ · (1/x) = eˣ(ln(x) + 1/x)
Mathematics is the queen of the sciences and number theory is the queen of mathematics.
— Carl Friedrich Gauss (1777–1855)
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What is the product rule?
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