Module 2: The Derivative
Calculus I – Study Guide
1. Rates of Change
Average Rate of Change = [f(b) − f(a)] / (b − a) = slope of secant line
Instantaneous Rate of Change = limh→0 [f(a+h) − f(a)] / h = slope of tangent line
2. Derivative Definition
f'(x) = limh→0 [f(x+h) − f(x)] / h
Alternative: f'(a) = limx→a [f(x) − f(a)] / (x − a)
Notation
f'(x) = dy/dx = df/dx = Df(x) — all mean the same thing.
3. Differentiability
Differentiable ⇒ Continuous (but NOT the other way around!)
Differentiability Fails At:
- Corners/cusps (left and right derivatives differ)
- Vertical tangents (slope = ∞)
- Discontinuities
4. Basic Differentiation Rules
| Rule | Formula |
|---|---|
| Constant | d/dx [c] = 0 |
| Power | d/dx [xn] = nxn−1 |
| Constant Multiple | d/dx [cf] = c · f' |
| Sum/Difference | d/dx [f ± g] = f' ± g' |
| Exponential | d/dx [ex] = ex |
| Natural Log | d/dx [ln x] = 1/x |
5. Tangent Line Formula
y − f(a) = f'(a)(x − a)
6. Higher Derivatives
f''(x) = second derivative (acceleration if f is position).
f(x) = x4: f' = 4x³, f'' = 12x², f''' = 24x, f(4) = 24.