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Differentiation – The quotient rule

📘 Notes

Differentiation — The Quotient Rule

The quotient rule is used when differentiating one function divided by another.

1. When do we use the quotient rule?

Use the quotient rule when the function is written as a fraction:

\[ y=\frac{u}{v} \]

Examples:

  • \(\dfrac{x^2+1}{x-3}\)
  • \(\dfrac{\sin x}{x}\)
  • \(\dfrac{e^x}{x^2+1}\)

2. The quotient rule formula

If \[ y=\frac{u}{v}, \] then

\[ \frac{dy}{dx}=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2} \]

3. Easy way to remember it

A common memory aid is:

\[ \frac{\text{bottom}\times\text{derivative of top}- \text{top}\times\text{derivative of bottom}} {(\text{bottom})^2} \]

Notice the order: bottom times derivative of top, minus top times derivative of bottom.

4. Worked example 1

Differentiate

\[ y=\frac{x^2+1}{x-3} \]

Let

\[ u=x^2+1,\qquad v=x-3 \]

Then

\[ \frac{du}{dx}=2x,\qquad \frac{dv}{dx}=1 \]

Using the quotient rule:

\[ \frac{dy}{dx} =\frac{(x-3)(2x)-(x^2+1)(1)}{(x-3)^2} \]

\[ =\frac{2x^2-6x-x^2-1}{(x-3)^2} \]

\[ \frac{dy}{dx}=\frac{x^2-6x-1}{(x-3)^2} \]

5. Worked example 2

Differentiate

\[ y=\frac{\sin x}{x} \]

Let

\[ u=\sin x,\qquad v=x \]

Then

\[ \frac{du}{dx}=\cos x,\qquad \frac{dv}{dx}=1 \]

So

\[ \frac{dy}{dx}=\frac{x\cos x-\sin x}{x^2} \]

\[ \frac{dy}{dx}=\frac{x\cos x-\sin x}{x^2} \]

6. Worked example 3

Differentiate

\[ y=\frac{e^x}{x^2+1} \]

Let

\[ u=e^x,\qquad v=x^2+1 \]

Then

\[ \frac{du}{dx}=e^x,\qquad \frac{dv}{dx}=2x \]

So

\[ \frac{dy}{dx} =\frac{(x^2+1)e^x-e^x(2x)}{(x^2+1)^2} \]

Factorising:

\[ \frac{dy}{dx}=\frac{e^x(x^2-2x+1)}{(x^2+1)^2} \]

7. When not to use the quotient rule

Sometimes it is easier to rewrite the fraction first.

Example:

\[ y=\frac{x^2+3x}{x}=x+3 \]

It is easier to simplify first, then differentiate: \[ \frac{dy}{dx}=1 \]

8. Quotient rule with chain rule

Sometimes one part of the fraction also needs the chain rule.

Example:

\[ y=\frac{1}{(2x+1)^3} \]

This can be done by quotient rule, but it is usually easier to rewrite first: \[ y=(2x+1)^{-3} \] and then use the chain rule.

9. Common mistakes

  • Forgetting to square the denominator.
  • Getting the order wrong in the numerator.
  • Forgetting the minus sign.
  • Not simplifying the final answer when possible.

10. Exam tips

  • Clearly identify the top and bottom as \(u\) and \(v\).
  • Write down \(u'\) and \(v'\) before substituting into the formula.
  • Use \[ \frac{dy}{dx}=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2} \] carefully.
  • Check whether simplifying first is easier than using the quotient rule directly.
  • Look out for chain rule inside the numerator or denominator.