1. Identify the Power ($n$): Look at the exponent of the variable.
2. Drop the Power: Bring $n$ to the front as a multiplier.
3. Subtract One: Reduce the old exponent by exactly 1 ($n-1$).
4. Linear Law: The derivative of $x$ is 1. The derivative of a constant is 0.
Find the derivative ($f'$) for each function using the Power Rule.
$f(x) = x^3$
$f(x) = x^{10}$
$f(x) = x^{100}$
Remember: Multiply the power by the number already in front.
$f(x) = 5x^2$
$f(x) = -2x^4$
$f(x) = 10x - 50$
If $f(x) = 7$, what is the derivative? Explain why this makes sense when you think about a car that is parked at mile marker 7.
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Apply the Power Rule to every term in the expression.
$f(x) = x^3 + 4x^2 - 7x + 12$
$f(x) = 2x^5 - \frac{1}{2}x^2 + 100$
1. The Root: Find the derivative of $f(x) = \sqrt{x}$. (Hint: write as $x^{1/2}$).
2. The Inverse: Find the derivative of $f(x) = 1/x$. (Hint: write as $x^{-1}$).
Objective: Explain the Power Rule to a younger student using their height.
The Activity:
1. Mark their height today.
2. Tell them that their "Height" is like a big number ($x^2$).
3. Tell them that how fast they grow is always a smaller number ($2x$).
The Lesson: "In God's world, things that are big move according to rules that are simple. The bigger you are, the more your speed depends on staying humble (subtracting 1)."
Response: ___________________________________________________________