UNIT 2: Differentiation: Definition and Basic Derivative Rules

Tips On Solving

+ After inputting the the equation, simplify it down so that the h = 0 is not at the bottom + Find the c and the equation and check f(x) spot to see if your right.

Derivatives That Fail To Exists

1. The derivative of a function, f(x), at a point where f(x) has a sharp turn or cusp does not exist. 2. The derivative of a function, f(x), at a point where f(x) has a vertical tangent line does not exist.

Tangent Line

A line in the plane of a circle that intersects the circle in exactly one point.

THEOREM: Equation of a Tangent Line

If M tan exists, then the equation of the tangent line to the graph of f at the point. (c, f(c)) is

Definition Of Tangent line With Slope M

If f is defined on an open interval containing c, and if the limit (equation) exists, then the line passing through the point (c, f(c)) with slope m is the tangent line to the graph of f at the point (c, f(c)) - materials from unit 1 and 2 combined

THEOREM: Differentiability Implies Continuity

If f is differentiable at x = c, then f is continuous at x = c. - The vice verse is not true/ It can be continuous but not differentiable

Alternate Form of Derivative / Definition

The derivative function f' of a function f is f'(x) =

Secant Line

The slope of this line will help you get tangent line

Prime Notation

f'(c) read "f prime of c" is used to denote the slope of a tangent line to a function f(c) at x=c. Therefore

Horizontal Tangent Line

f'(x)=0