Summary AP Calculus AB Unit 2: Differentiation: Definition and Fundamental Properties - Study Guide

AP Calculus AB Unit 2: Differentiation



Differentiation
Definition and Fundamental Properties




1. Definition of the Derivative

The derivative represents the instantaneous rate of change of a function, or the slope of the tangent line
at a specific point.

Limit Definition of the Derivative
The derivative of f at x, denoted by f ′ (x), is defined as:

f (x + h) − f (x)
f ′ (x) = lim
h→0 h
Alternative Form (Derivative at a point x = a):

f (x) − f (a)
f ′ (a) = lim
x→a x−a

Notation for Derivative: • Lagrange: f ′ (x) or y ′
dy
• Leibniz: dx or d
dx [f (x)]



2. Differentiability and Continuity

Theorem: If a function is differentiable at x = c, it must be continuous at x = c.
Note: The converse is false. Continuity does not guarantee differentiability.

Where Derivatives Fail (Non-Differentiable Points):

1. Corner/Cusp: Sharp turn (slopes from left and right differ). Example: y = |x| at x = 0.
√
2. Vertical Tangent: Slope is undefined (∞). Example: y = 3 x at x = 0.

3. Discontinuity: Any hole, jump, or asymptote.


3. Power and Sum/Difference Rules

• Power Rule:
d n
(x ) = nxn−1
dx
• Constant Rule: d
dx (c) =0

• Constant Multiple Rule: d
dx [c · f (x)] = c · f ′ (x)


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