Quotient Rule

TL;DR

The Quotient Rule helps you find the derivative of a function that's a fraction of two other functions. It's a specific formula you apply when you can't simplify the fraction first. Think of it as a special tool for "top-over-bottom" derivatives.

1. The Mental Model

Imagine you have a cake (a function) that's made up of two distinct layers: a top layer and a bottom layer, and you need to find out how quickly the overall cake's "value" changes. The Quotient Rule gives you a systematic way to combine the changes in each layer to figure out the overall change.

2. The Core Material

The Quotient Rule is how you find the derivative of a function $f(x)$ that looks like a fraction, $f(x) = \frac{g(x)}{h(x)}$, where both $g(x)$ and $h(x)$ are differentiable functions.

The formula is:

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

Let's break down what each part means:

• $g(x)$: This is your "top" function (the numerator).
• $h(x)$: This is your "bottom" function (the denominator).
• $g'(x)$: This is the derivative of your top function.
• $h'(x)$: This is the derivative of your bottom function.
• $[h(x)]^2$: This is the square of your original bottom function.

A common way to remember the formula is: "low d-high minus high d-low, all over low squared."
* "low" refers to $h(x)$
* "d-high" refers to $g'(x)$ (derivative of high)
* "high" refers to $g(x)$
* "d-low" refers to $h'(x)$ (derivative of low)
* "low squared" refers to $[h(x)]^2$

Steps to Apply the Quotient Rule

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1. Identify $g(x)$ and $h(x)$: Clearly define your numerator and denominator functions.
2. Find their derivatives: Calculate $g'(x)$ and $h'(x)$.
3. Plug into the formula: Substitute $g(x)$, $h(x)$, $g'(x)$, and $h'(x)$ into the Quotient Rule formula.
4. Simplify: Algebraically simplify the resulting expression as much as possible.

graph TD
    A["Start: You have f(x) = g(x)/h(x)"] --> B["Identify g(x) (numerator)"]
    A --> C["Identify h(x) (denominator)"]
    B --> D["Calculate g'(x)"]
    C --> E["Calculate h'(x)"]
    D & E --> F["Apply Formula: (g'(x)h(x) - g(x)h'(x)) / [h(x)]^2"]
    F --> G["Simplify the Result"]
    G --> H["End: Derivative is found"]

3. Worked Example

Let's find the derivative of $f(x) = \frac{x^2 + 1}{2x - 3}$.

1. Identify $g(x)$ and $h(x)$:

   • $g(x) = x^2 + 1$ (the top function)
   • $h(x) = 2x - 3$ (the bottom function)

2. Find their derivatives:

   • $g'(x) = \frac{d}{dx}(x^2 + 1) = 2x$
   • $h'(x) = \frac{d}{dx}(2x - 3) = 2$

3. Plug into the formula:
   Recall the formula: $f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2}$

   Substitute the functions and their derivatives:
   $f'(x) = \frac{(2x)(2x - 3) - (x^2 + 1)(2)}{(2x - 3)^2}$

4. Simplify:
   Expand the terms in the numerator:
   $f'(x) = \frac{(4x^2 - 6x) - (2x^2 + 2)}{(2x - 3)^2}$

   Distribute the negative sign:
   $f'(x) = \frac{4x^2 - 6x - 2x^2 - 2}{(2x - 3)^2}$

   Combine like terms in the numerator:
   $f'(x) = \frac{2x^2 - 6x - 2}{(2x - 3)^2}$

This is your final derivative. You could factor a 2 from the numerator, but it doesn't simplify further with the denominator, so this form is perfectly acceptable.

4. Key Takeaways

• The Quotient Rule is for differentiating functions that are fractions: $f(x) = \frac{g(x)}{h(x)}$.
• The formula is $\frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2}$, often remembered as "low d-high minus high d-low, all over low squared."
• Always identify your "top" and "bottom" functions and their derivatives before plugging into the formula.
• The order of terms in the numerator matters due to the subtraction: $g'(x)h(x) - g(x)h'(x)$ is not the same as $g(x)h'(x) - g'(x)h(x)$.
• Don't forget to square the denominator in the final step of the formula.

Common Mistakes to Avoid:

• Forgetting the subtraction order: Make sure it's (derivative of top * bottom) minus (top * derivative of bottom).
• Not squaring the denominator: It's $[h(x)]^2$, not just $h(x)$.
• Incorrectly differentiating $g(x)$ or $h(x)$: Make sure your initial derivative calculations are correct.
• Trying to use the Product Rule: The Product Rule is for multiplication, not division.
• Not simplifying the numerator correctly: Be careful with distributing negative signs and combining like terms.

5. Now Try It

Find the derivative of the function $y = \frac{5x}{x^2 + 4}$.

Your steps should include:
1. Clearly identifying $g(x)$ and $h(x)$.
2. Calculating $g'(x)$ and $h'(x)$.
3. Plugging these into the Quotient Rule formula.
4. Simplifying the resulting expression to its most compact form.

Success looks like a final derivative expression with a simplified numerator and the squared denominator.