The product and quotient rules

Covers AS91578

The product rule

Use the product rule when two functions are multiplied together, like $x^2 e^x$ .
If $y = uv$ (with $u$ and $v$ both functions of $x$ ):

$\frac{dy}{dx} = u'v + uv'$

In words: derivative of the first times the second, plus the first times the derivative of the second.

The quotient rule

Use the quotient rule when one function is divided by another, like $\dfrac{x}{x + 1}$ .
If $y = \dfrac{u}{v}$ :

$\frac{dy}{dx} = \frac{u'v - uv'}{v^2}$

The order matters here (it is a subtraction), and the whole thing is divided by $v^2$ .

Worked ExampleProduct rule

Differentiate $y = x^2 e^x$ .

Step 1 — Identify the two factors and their derivatives

$u = x^2 \Rightarrow u' = 2x \qquad v = e^x \Rightarrow v' = e^x$

Step 2 — Apply $u'v + uv'$

$\frac{dy}{dx} = (2x)(e^x) + (x^2)(e^x)$

Step 3 — Factorise to tidy the answer

$\frac{dy}{dx} = e^x(2x + x^2)$

Worked ExampleQuotient rule

Differentiate $y = \dfrac{x}{x + 1}$ .

Step 1 — Identify $u$ , $v$ and their derivatives

$u = x \Rightarrow u' = 1 \qquad v = x + 1 \Rightarrow v' = 1$

Step 2 — Apply $\dfrac{u'v - uv'}{v^2}$

$\frac{dy}{dx} = \frac{(1)(x + 1) - (x)(1)}{(x + 1)^2}$

Step 3 — Simplify the numerator

$= \frac{x + 1 - x}{(x + 1)^2} = \frac{1}{(x + 1)^2}$

Test yourself

Practice by grade

One question each at Achieved, Merit and Excellence. Have a go, then compare with the model answer.

Achieved

Differentiate $y = x^3(x + 2)$ using the product rule.

Merit

Differentiate $y = \dfrac{2x + 1}{x - 3}$ .

Excellence

Differentiate $y = x^2\sin x$ and find the gradient at $x = 0$ .

The product and quotient rules

Covers AS91578

The product rule

Use the product rule when two functions are multiplied together, like $x^2 e^x$ .
If $y = uv$ (with $u$ and $v$ both functions of $x$ ):

$\frac{dy}{dx} = u'v + uv'$

In words: derivative of the first times the second, plus the first times the derivative of the second.

The quotient rule

Use the quotient rule when one function is divided by another, like $\dfrac{x}{x + 1}$ .
If $y = \dfrac{u}{v}$ :

$\frac{dy}{dx} = \frac{u'v - uv'}{v^2}$

The order matters here (it is a subtraction), and the whole thing is divided by $v^2$ .

Worked ExampleProduct rule

Differentiate $y = x^2 e^x$ .

Step 1 — Identify the two factors and their derivatives

$u = x^2 \Rightarrow u' = 2x \qquad v = e^x \Rightarrow v' = e^x$

Step 2 — Apply $u'v + uv'$

$\frac{dy}{dx} = (2x)(e^x) + (x^2)(e^x)$

Step 3 — Factorise to tidy the answer

$\frac{dy}{dx} = e^x(2x + x^2)$

Worked ExampleQuotient rule

Differentiate $y = \dfrac{x}{x + 1}$ .

Step 1 — Identify $u$ , $v$ and their derivatives

$u = x \Rightarrow u' = 1 \qquad v = x + 1 \Rightarrow v' = 1$

Step 2 — Apply $\dfrac{u'v - uv'}{v^2}$

$\frac{dy}{dx} = \frac{(1)(x + 1) - (x)(1)}{(x + 1)^2}$

Step 3 — Simplify the numerator

$= \frac{x + 1 - x}{(x + 1)^2} = \frac{1}{(x + 1)^2}$

Test yourself

Practice by grade

One question each at Achieved, Merit and Excellence. Have a go, then compare with the model answer.

Achieved

Differentiate $y = x^3(x + 2)$ using the product rule.

Merit

Differentiate $y = \dfrac{2x + 1}{x - 3}$ .

Excellence

Differentiate $y = x^2\sin x$ and find the gradient at $x = 0$ .

The product and quotient rules

The product rule

The quotient rule

Step 1 — Identify the two factors and their derivatives

Step 2 — Apply u′v+uv′u'v + uv'u′v+uv′

Step 3 — Factorise to tidy the answer

Step 1 — Identify uuu, vvv and their derivatives

Step 2 — Apply u′v−uv′v2\dfrac{u'v - uv'}{v^2}v2u′v−uv′​

Step 3 — Simplify the numerator

Test yourself

Practice by grade

The product and quotient rules

The product rule

The quotient rule

Step 1 — Identify the two factors and their derivatives

Step 2 — Apply u′v+uv′u'v + uv'u′v+uv′

Step 3 — Factorise to tidy the answer

Step 1 — Identify uuu, vvv and their derivatives

Step 2 — Apply u′v−uv′v2\dfrac{u'v - uv'}{v^2}v2u′v−uv′​

Step 3 — Simplify the numerator

Test yourself

Practice by grade

Step 2 — Apply $u'v + uv'$

Step 1 — Identify $u$ , $v$ and their derivatives

Step 2 — Apply $\dfrac{u'v - uv'}{v^2}$

Step 2 — Apply $u'v + uv'$

Step 1 — Identify $u$ , $v$ and their derivatives

Step 2 — Apply $\dfrac{u'v - uv'}{v^2}$