Module 2 Quick Reference

The Derivative – Calculus I

Derivative Definition

f'(x) = limh→0 [f(x+h) − f(x)] / h
f'(a) = limx→a [f(x) − f(a)] / (x − a)

Basic Rules

Power Rule

d/dx [xn] = nxn−1

Constant

d/dx [c] = 0

Sum/Diff

(f ± g)' = f' ± g'

Const Multiple

(cf)' = cf'

Special Derivatives

d/dx [ex] = ex

d/dx [ln x] = 1/x

Tangent Line

y − f(a) = f'(a)(x − a)

Differentiability

Differentiable ⇒ Continuous (not converse)

Fails at: corners, cusps, vertical tangents, discontinuities

Interpretations

Geometric

Slope of tangent line

Physical

Instantaneous rate of change