Product & Quotient Rules: Managing Relationships Between Functions

These rules allow us to differentiate expressions where two functions are either multiplied or divided. While the Power Rule works for individual terms, these rules are essential for handling more complex combinations.

Core Theorem

**Product Rule:**

\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)

**Quotient Rule:**

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

TL;DR: Product: $u'v + uv'$. Quotient: **"Low d-High minus High d-Low, over Low-Low."**

Practice Exercises

Example 01Medium

Find

y'

for

y = x^2 \ln(x)

NEED A HINT?

This is a product of

f(x)=x^2

and

g(x)=\ln(x)

. Use the Product Rule.

SHOW DETAILED EXPLANATION

Step 1: Identify Parts

u = x^2, v = \ln(x)

Step 2: Differentiate Each

u' = 2x, v' = \frac{1}{x}

Step 3: Apply Rule

y' = (2x)(\ln x) + (x^2)(\frac{1}{x}) = 2x\ln x + x

Example 02AB/BC Standard

Differentiate

f(x) = \frac{\sin(x)}{x^2}

NEED A HINT?

Use the Quotient Rule:

\frac{Lo \cdot dHi - Hi \cdot dLo}{Lo^2}

SHOW DETAILED EXPLANATION

Step 1: Identify Low and High

High = \sin(x), Low = x^2

Step 2: Differentiate Each

dHigh = \cos(x), dLow = 2x

Step 3: Assemble

f'(x) = \frac{(x^2)(\cos x) - (\sin x)(2x)}{(x^2)^2} = \frac{x\cos x - 2\sin x}{x^3}

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