How to differentiate using the quotient rule

In Add Math, many expressions are one function divided by another, such as $y=\frac{2x+1}{x-3}$, $y=\frac{x^{2}}{2x+1}$ or $y=\frac{3x}{x^{2}+1}$. The quotient rule is the tool for differentiating a fraction where both the numerator and the denominator contain $x$.

You cannot simply differentiate the top and bottom separately, and dividing the two derivatives is wrong. The quotient rule gives a single reliable formula that keeps track of both parts and, crucially, the order in which they are combined.

It appears throughout the Differentiation chapter and is exactly what you need for gradients, tangents, normals and stationary points of curves written as fractions. A neat, well-set-out application also earns full method marks, which the analytic marking scheme rewards.

Reach for the quotient rule whenever $y$ is written as a fraction and both the numerator and denominator are functions of $x$, such as $\frac{x+2}{2x-1}$ or $\frac{x^{2}-1}{x+4}$. The dividing line is the signal.

One check first: if the denominator is just a constant, you do not need the rule, $\frac{2x+1}{5}$ is simply $\tfrac{1}{5}(2x+1)$. And if the fraction can be split into simple powers, for example $\frac{x^{2}+x}{x}=x+1$, simplify first and differentiate directly.

Use the quotient rule when the fraction genuinely cannot be simplified away. Tell it apart from the other rules: quotient for division, product for multiplication, and chain for one function wrapped inside another.

1. 1

   Name top and bottom

   Let $u$ be the numerator and $v$ be the denominator, so $y=\frac{u}{v}$.

2. 2

   Differentiate each part

   Find $\frac{du}{dx}$ and $\frac{dv}{dx}$ separately.

3. 3

   Build the numerator

   Write $v\frac{du}{dx}-u\frac{dv}{dx}$. The order matters: it is bottom times derivative of top, minus top times derivative of bottom.

4. 4

   Square the denominator

   The whole thing sits over $v^{2}$.

5. 5

   Simplify the top

   Expand and collect like terms in the numerator; leave the denominator as $v^{2}$.

6. 6

   Use it if a value is asked

   Substitute the given $x$-value for a gradient, or set the numerator equal to $0$ for a stationary point.

• Getting the order wrong in the numerator: it is $v\frac{du}{dx}-u\frac{dv}{dx}$, not $u\frac{dv}{dx}-v\frac{du}{dx}$. The wrong order flips the sign of the whole answer.
• Forgetting to square the denominator, or writing $v$ instead of $v^{2}$.
• Dropping the bracket when subtracting, so $-(2x+1)$ becomes $-2x+1$ instead of $-2x-1$.
• Trying to differentiate the top and bottom separately and divide, the quotient rule does not work that way.
• Not simplifying, or cancelling terms inside the squared denominator that cannot be cancelled.

The step students most often get wrong is the order and the subtraction sign in the numerator, one small slip turns $-7$ into $+7$ and loses the marks. In a one-to-one lesson our teachers get you to write $u$, $v$ and their derivatives first, then say the rule aloud as 'bottom d-top minus top d-bottom, all over bottom squared', so the order is locked in.

Because your working is shown line by line, the bracket in the subtraction stays in place and every method mark is easy to award. Teachers at spmaddmath.com.my are experienced, and lessons are online and taught in English.

To see how we teach the quotient rule, message us on WhatsApp to arrange a one-hour paid class from RM50/hr at the teacher's rate.