The Quotient Rule

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

In Words

Bottom times derivative of top, MINUS top times derivative of bottom, ALL OVER bottom squared

$\frac{[\text{bottom}] \cdot [\text{top}]' - [\text{top}] \cdot [\text{bottom}]'}{[\text{bottom}]^2}$

Memory Tricks

"Low dee-high minus high dee-low, over low-low"

Or sing it to a tune:

♪ Low dee-high,
  Minus high dee-low,
  All over low squared! ♪

Visual:

    f
   ---
    g

becomes

  g·f' - f·g'
  -----------
      g²

Basic Example

Find $\frac{d}{dx}\left[\frac{x^2}{x + 1}\right]$

Step 1: Identify top and bottom

• Top (numerator): $f(x) = x^2$, so $f'(x) = 2x$
• Bottom (denominator): $g(x) = x + 1$, so $g'(x) = 1$

Step 2: Apply quotient rule $= \frac{g(x) \cdot f'(x) - f(x) \cdot g'(x)}{[g(x)]^2}$

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

Step 3: Expand numerator $= \frac{2x^2 + 2x - x^2}{(x + 1)^2}$

Step 4: Simplify $= \frac{x^2 + 2x}{(x + 1)^2}$

We can factor if desired: $= \frac{x(x + 2)}{(x + 1)^2}$

Important: Order Matters!

The quotient rule has a minus sign, so order is crucial!

$g \cdot f' - f \cdot g' \quad \neq \quad f \cdot g' - g \cdot f'$

Always: bottom times top prime minus top times bottom prime

When Can You Avoid the Quotient Rule?

Case 1: Constant in numerator

$\frac{d}{dx}\left[\frac{5}{x^2}\right] = \frac{d}{dx}[5x^{-2}] = -10x^{-3} = -\frac{10}{x^3}$

Use negative exponent instead!

Case 2: Factorable expression

Sometimes you can simplify before differentiating.

Step-by-Step Example

Find $\frac{d}{dx}\left[\frac{3x + 2}{x^2 - 1}\right]$

Step 1: Identify

• Top: $f(x) = 3x + 2$, so $f'(x) = 3$
• Bottom: $g(x) = x^2 - 1$, so $g'(x) = 2x$

Step 2: Write the quotient rule $= \frac{g \cdot f' - f \cdot g'}{g^2}$

Step 3: Substitute $= \frac{(x^2 - 1)(3) - (3x + 2)(2x)}{(x^2 - 1)^2}$

Step 4: Expand numerator $= \frac{3x^2 - 3 - (6x^2 + 4x)}{(x^2 - 1)^2}$

$= \frac{3x^2 - 3 - 6x^2 - 4x}{(x^2 - 1)^2}$

Step 5: Combine like terms $= \frac{-3x^2 - 4x - 3}{(x^2 - 1)^2}$

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

Quotient Rule vs. Rewriting

Sometimes rewriting is easier:

Hard way (quotient rule): $\frac{d}{dx}\left[\frac{1}{x^3}\right]$

Easy way (power rule): $\frac{d}{dx}[x^{-3}] = -3x^{-4} = -\frac{3}{x^4}$

If the numerator is constant or simple, consider rewriting!

Product Rule + Chain Rule Alternative

You can also rewrite quotients as products:

$\frac{f(x)}{g(x)} = f(x) \cdot [g(x)]^{-1}$

Then use product rule + chain rule. But quotient rule is usually faster!

Common Mistakes

❌ Wrong sign: $g \cdot f' + f \cdot g'$ (should be minus!)

❌ Forgetting to square bottom: $\frac{...}{g}$ (should be $g^2$!)

❌ Wrong order: $f \cdot g' - g \cdot f'$ (backwards!)

✓ Correct: $\frac{g \cdot f' - f \cdot g'}{g^2}$

Simplification Tips

After applying the quotient rule:

1. Expand the numerator completely
2. Combine like terms
3. Factor if possible
4. Don't expand $(g(x))^2$ unless necessary

Practice Strategy

1. Circle the numerator (top)
2. Box the denominator (bottom)
3. Find both derivatives
4. Write the quotient rule formula
5. Substitute carefully (watch the minus!)
6. Simplify the numerator
7. Leave denominator factored (usually)

Step 2: Apply quotient rule $\frac{dy}{dx} = \frac{g \cdot f' - f \cdot g'}{g^2}$

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

Step 3: Expand numerator $= \frac{x - 2 - x - 3}{(x - 2)^2}$

Step 4: Simplify $= \frac{-5}{(x - 2)^2}$

Answer: dy/dx = -5/(x - 2)²

Let $u = x^2 + 3x$ and $v = x - 2$

$u' = 2x + 3$ $v' = 1$

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

Expand numerator:

$= \frac{2x^2 - 4x + 3x - 6 - x^2 - 3x}{(x - 2)^2}$

$= \frac{2x^2 - x^2 - 4x + 3x - 3x - 6}{(x - 2)^2}$

$= \frac{x^2 - 4x - 6}{(x - 2)^2}$

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

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

Use Pythagorean identity: $\sin^2 x + \cos^2 x = 1$

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

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

$= \frac{-1}{1 + \sin x}$