1. Substitute: Replace every $x$ in the function with $(x+h)$.
2. Expand: Multiply everything out. Be careful with signs!
3. Subtract: Subtract the original $f(x)$. All terms without $h$ should vanish.
4. Divide: Cancel the $h$ on top with the $h$ on bottom.
5. Limit: Set remaining $h$'s to 0.
Find the derivative of $f(x) = 4x + 2$ using the definition.
Step 1: Find $f(x+h)$.
Step 2 & 3: Subtract $f(x)$ and divide by $h$.
Step 4: What is the limit as $h o 0$?
Find the derivative of $f(x) = x^2 + 3x$.
The Setup: Write out the Difference Quotient.
The Expansion: Expand $(x+h)^2$.
The Cleanup: Cancel terms and the $h$. What is left?
In Part II, after you cancel the $h$, you should have $2x + h + 3$. When you let $h o 0$, what happens to the $+ h$? Does it become 1 or 0?
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Find the derivative of $f(x) = x^3$.
Hint: $(x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3$.
Objective: Explain "Closing the Gap" to a younger student.
The Activity:
1. Draw two dots on a paper far apart.
2. Draw a line connecting them.
3. Ask: "If I move the dots closer, does the line change direction?" (Yes).
4. "If the dots touch, the line points exactly where they are going."
The Lesson: "We want to be so close to God (no gap) that His direction is our direction."
Response: ___________________________________________________________