Topic 2.9 Notes – The Quotient Rule

1. What the Quotient Rule Is

If $f$ and $g$ are differentiable and $g(x) \ne 0$, then

$$\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) = \frac{g(x)f'(x) - f(x)g'(x)}{(g(x))^2}$$

You’ll also see it written as

$$\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{vu' - uv'}{v^2}$$

The structure matters:

• Bottom × derivative of top
• Minus
• Top × derivative of bottom
• All over bottom squared

A quick memory phrase many students use:
“Low d-high minus high d-low, over low squared.”

Two conditions:

• Both numerator and denominator must be differentiable.
• The denominator cannot equal zero.

It feels similar to the product rule, except instead of adding two products, you subtract and divide by the square of the denominator.

2. When to Use It (and When Not To)

The Quotient Rule is for expressions that are truly quotients.

Use it when:

• The function is clearly $\frac{f(x)}{g(x)}$.
• The denominator isn’t a constant.
• Rewriting would make things messier.

Typical cases:

• Rational functions like $\frac{x^3 - 1}{x^2 + 4}$
• Trig over trig
• Exponential over polynomial

But sometimes rewriting is faster.

For example:

• $\frac{5x^4}{2} = \frac{5}{2}x^4$ → just use power rule.
• $\frac{1}{x^6} = x^{-6}$ → power rule again.

On no-calculator MCQs, rewriting often saves time and reduces sign mistakes.

3. How to Apply the Quotient Rule

Take
$$y = \frac{x^3 + 2x}{x^2 - 1}$$

Step 1: Identify pieces

• $f(x) = x^3 + 2x$
• $g(x) = x^2 - 1$

Step 2: Differentiate each

• $f'(x) = 3x^2 + 2$
• $g'(x) = 2x$

Step 3: Plug into formula

$$y' = \frac{(x^2 - 1)(3x^2 + 2) - (x^3 + 2x)(2x)}{(x^2 - 1)^2}$$

Now simplify the numerator carefully:

$$= \frac{(3x^4 + 2x^2 - 3x^2 - 2) - (2x^4 + 4x^2)}{(x^2 - 1)^2}$$

$$= \frac{x^4 - 5x^2 - 2}{(x^2 - 1)^2}$$

Notice:

• Parentheses prevented sign errors.
• The denominator stayed squared.
• We simplified only after plugging everything in.

That algebra cleanup step is where most quiz mistakes happen.

4. Special Situations You’ll See on Tests

A. Exponentials

Example structure:
$$\frac{e^{2x}}{x^3 + 1}$$

You still follow the same rule. Just remember:
$$\frac{d}{dx}(e^{2x}) = 2e^{2x}$$

Chain rule happens first inside the numerator, then the quotient rule.

Factoring out $e^{2x}$ afterward often simplifies things.

B. Trigonometric Functions

You must know:
$$\frac{d}{dx}(\sin x) = \cos x$$ $$\frac{d}{dx}(\cos x) = -\sin x$$

Example structure:
$$\frac{\tan x}{x}$$

Differentiate top and bottom separately, then apply the rule.

Afterward, look for trig identities to simplify. On FRQs, clean simplification can make later parts easier.

C. Chain Rule + Quotient Rule Together

Something like:
$$\frac{\ln(3x)}{x^2 + 5}$$

You first differentiate $\ln(3x)$ using chain rule:
$$\frac{d}{dx}[\ln(3x)] = \frac{3}{3x} = \frac{1}{x}$$

Then plug into the quotient rule.

When problems combine rules, the order is always:

1. Differentiate inner pieces correctly.
2. Substitute into the quotient structure.
3. Simplify.

5. Common Mistakes

These show up constantly on tests:

• Dropping the minus sign
  It’s $gf' - fg'$, not plus.
• Forgetting to square the denominator
  The entire denominator gets squared, even if it’s multiple terms.
• Losing parentheses
  Especially when subtracting the second product.
• Using product rule by accident
  If you’re adding instead of subtracting, that’s wrong.
• Expanding too early
  Keep structure clean, then simplify.