VG3
kalkulus
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)
🧠 Test yourself
Question 1 of 5
What is the product rule?
Read also
What is the quotient rule?
The quotient rule is used when differentiating a fraction of two functions.…
Completing the square
Completing the square is an elegant technique for solving quadratic equations and rewritin…
What is the indefinite integral?
The indefinite integral is the reverse of differentiation — it finds the antiderivative F(…