The product and quotient rules differentiate functions formed by multiplying or dividing two differentiable functions.

What this dot point is asking

You need to recognise when a function is a product of two functions or a quotient of two functions, choose the correct rule, and apply it carefully. These rules build on the basic derivatives of polynomials, exponentials, logarithms and trig functions, so accuracy in those is assumed.

The product rule

If $f(x)=u(x)\,v(x)$ where $u$ and $v$ are both differentiable, then

A useful shorthand is "first times derivative of second, plus second times derivative of first" - but written as $u'v+uv'$ the order does not matter because addition is commutative.

When to use it

Use the product rule whenever the function is a multiplication of two non-trivial expressions, for example $x^2 e^x$, $x\sin x$, or $(2x+1)\ln x$. If one factor is a constant you do not need the product rule - just keep the constant.

The quotient rule

If $f(x)=\dfrac{u(x)}{v(x)}$ where $u$ and $v$ are differentiable and $v(x)\neq 0$, then

The order in the numerator does matter here because of the subtraction: it is always (derivative of top)(bottom) minus (top)(derivative of bottom), all over (bottom) squared.

Choosing the right rule

• A product (two things multiplied): product rule.
• A quotient (one thing divided by another): quotient rule, or rewrite as a product with a negative power and use the product/chain rule.
• A sum or difference: differentiate term by term - no special rule needed.

Sometimes a quotient is easier as a product. For example $\dfrac{e^x}{x^2}=e^x x^{-2}$ can be differentiated with the product rule, which some students find less error-prone than the quotient rule.

Common errors

Bringing it together

The product and quotient rules are the workhorses of Stage 2 differentiation. They combine constantly with the chain rule (the next dot point) - for instance $\dfrac{d}{dx}\big[x e^{2x}\big]$ needs the product rule on the outside and the chain rule to differentiate $e^{2x}$. Master identifying $u$ and $v$ cleanly, write the rule before substituting, and factorise your answer where possible so it is ready for the applications that follow.