1. Label the Pair: Write $u = (...)$ and $v = (...)$ for every problem.
2. Differentiate Individually: Find $u'$ and $v'$ using the Power Rule.
3. Assemble the Union: Write out $u \cdot v' + v \cdot u'$.
4. The "Plus" Law: Products grow by addition of their interactive rates.
Find the derivative ($f'$) using the Product Rule. Show all four parts ($u, v, u', v'$).
$f(x) = (x^2)(x^4)$
$f(x) = (3x^2 + 5)(x^3)$
$f(x) = (x + 10)(x - 2)$
In the third problem, if you "FOILed" the expression first to get $x^2 + 8x - 20$ and then took the derivative, would you get the same answer? Check it!
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$f(x) = (x^2 + 4x + 1)(2x - 5)$
$f(x) = (\sqrt{x})(x^2 + 5)$
A ministry's influence is the product of its **Number of Leaders** ($L$) and its **Depth of Teaching** ($D$).
$L(t) = 5t + 20$ (Leaders growing over time).
$D(t) = t^2$ (Depth increasing as they study).
Total Influence $I(t) = L(t) \cdot D(t)$.
Task: Find the rate of increase of Influence ($I'$) at $t = 2$ years.
Objective: Explain the Product Rule to a younger student using Lego blocks.
The Activity:
1. Make two towers of different colors.
2. Tell them the "Power" of the towers is their heights multiplied.
3. Add one block to the first tower. Ask: "How much did the total area change?" (It's the height of the second tower).
4. Add one block to the second tower. Ask: "How much changed now?" (Height of the first tower).
The Lesson: "In a team, when I grow, I add your height to the team. When you grow, you add mine. We carry each other."
Response: ___________________________________________________________