1. Learn the Symbols: $f'(x)$ and $\frac{dy}{dx}$ mean the same thing (Slope).
2. Smoothness Rule: If there is a sharp point or a hole, you cannot take the derivative.
3. Slope Reading: Uphill = Positive (+). Downhill = Negative (-). Flat = Zero (0).
4. The Power Shortcut: Bring the exponent down, subtract 1.
Look at the graph provided (draw a wavy curve). Identify the sign of the derivative ($f'$) at different points.
Point A (Climbing the hill): Is $f'(A)$ positive, negative, or zero?
Point B (The Peak): Is $f'(B)$ positive, negative, or zero?
Point C (Sliding down): Is $f'(C)$ positive, negative, or zero?
Decide if the function has a derivative at $x=0$.
The Parabola: $y = x^2$. Is it smooth at the bottom? Can you draw a tangent?
The Absolute Value: $y = |x|$ (A V-shape). Is it smooth at the point? Can you draw a unique tangent?
If a car is driving North at 50mph, stops instantly (0 seconds), and drives South at 50mph... is that physically possible? Why does physics require "smoothness"?
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The Sine Wave:
Draw $y = \sin(x)$ (starts at 0, goes up).
Below it, sketch the slope.
At $x=0$, slope is +1.
At the peak, slope is 0.
Does your derivative graph look like $\cos(x)$?
Given position $s(t) = t^3$.
1. Find Velocity ($v = s'$).
2. Find Acceleration ($a = v' = s''$).
Objective: Teach a younger student to "feel" the slope.
The Activity:
1. Draw a roller coaster on paper.
2. Have them run their hand along it.
3. Ask: "Where is your hand flat? (Top/Bottom). Where is it pointing up? Where is it pointing down?"
The Lesson: "Math has a special machine called a Derivative that tells us exactly which way your hand is pointing at any spot on the ride."
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