The product rule

When two functions of x are multiplied, you need the product rule. The derivative is not simply the product of the derivatives.

Rules

$(u \cdot v)^{'} = u^{'} v + u v^{'}$
Write down $u$ , $u^{'}$ , $v$ , $v^{'}$ separately first, then put them in
Factor out common terms at the end (e.g. $e^{x}$ )

Your turn

Three problems on this topic. Type like on a calculator: 3x^2 - 2/x, e^(2x), sqrt(x), ln(x).

Task 1 of 3Product rule

Task

Differentiate:

f (x) = 5 x cos (x)

$f^{'} (x) =$

Write like on a calculator: 3x^2 - 2/x, e^(2x), sqrt(x), ln(x), sin(x). Use brackets: e^(2x), not e^2x. Numbers also as 3/4 or e^2 - 1.

Worked example

Differentiate: $f (x) = x^{2} e^{- x}$

Worked solution

1.Product rule: $(u v)^{'} = u^{'} v + u v^{'}$
2. $u = x^{2}, u^{'} = 2 x^{1}$ · $v = e^{- x}, v^{'} = - e^{- x}$ (chain rule)
3. $f^{'} (x) = 2 x e^{- x} - x^{2} e^{- x} = e^{- x} x^{1} (2 - x)$

Typical mistake: Computing $u^{'} \cdot v^{'}$ ; the derivative of a product is not the product of the derivatives.

Differentiate: $f (x) = 4 x sin (x)$

Worked solution

1.Product rule: $(u v)^{'} = u^{'} v + u v^{'}$
2. $u = 4 x, u^{'} = 4$ · $v = sin x, v^{'} = cos x$
3. $f^{'} (x) = 4 sin (x) + 4 x cos (x)$

Typical mistake: Forgetting the minus in $(cos x)^{'} = - sin x$ .

Common questions

When do I need the product rule?

When two parts that both depend on x are multiplied, e.g. x² · e^x or x · sin x. For a constant times a function the constant multiple rule is enough.

Chain rule Quotient rule Integration by parts

The product rule

Rules

Your turn

When do I need the product rule?

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