1. Label the High and Low: Top = $u$, Bottom = $v$.
2. Differentiate Individually: Find $u'$ and $v'$.
3. Follow the Mnemonic: Low-D-High ($v \cdot u'$) minus High-D-Low ($u \cdot v'$).
4. Square the Foundation: Put the result over $(v)^2$.
Find the derivative ($f'$) using the Quotient Rule. Show all parts ($u, v, u', v'$).
$f(x) = \frac{x^2}{x + 3}$
$f(x) = \frac{5x}{x^2 + 1}$
$f(x) = \frac{x^3}{x^2 - 4}$
Why do we use a Minus Sign in the Quotient Rule, but a Plus Sign in the Product Rule? What does this tell you about the difference between "multiplying" a relationship and "dividing" a responsibility?
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$f(x) = \frac{x^2 + 4x + 1}{2x - 5}$
$f(x) = \frac{\sqrt{x}}{x + 10}$
A city's "Order" ($O$) is defined as its **Knowledge** ($K$) divided by its **Confusion** ($C$).
$K(t) = 10t^2$ (Knowledge growing over time).
$C(t) = 2t + 5$ (Confusion also growing as the city gets bigger).
Total Order $O(t) = K(t) / C(t)$.
Task: Find the rate of change of Order ($O'$) at $t = 5$ years.
Objective: Explain the Quotient Rule to a younger student using a piggyback ride.
The Activity:
1. Have an older student carry a younger student.
2. Have the younger student "reach up" for a toy (change in $u$).
3. Have the older student "stumble" (change in $v$).
The Lesson: "In a team where one carries another, when the bottom person stumbles, it hurts the whole team. But when the bottom person stays strong (squared), the top person can reach higher!"
Response: ___________________________________________________________