SECTION 3.1The Derivative and the Tangent Line Problem

Remember what the notion of limits allows us to do . . .

Tangency

Instantaneous Rate of Change

The Notion of a DerivativeDerivative• The instantaneous rate of change of a function.• Think “slope of the tangent line.”

Definition of the Derivative of a Function (p. 119)The derivative of at is given by

Provided the limit exists. For all for which this limit exists, is a function of .

Graphical Representation

f(x)

So, what’s the point?

f(x)

f(x)

f(x)

Notation and Terminology

Terminology

differentiation, differentiable, differentiable on an open interval (a,b)

Differing Notation Representing “Derivative”

Example 1 (#2b)Estimate the slope of the graph at the points and .

Example 2Find the derivative by the limit process (a.k.a. the formal definition).

a.

b.

Example 3Find an equation of the tangent line to the graph of at the given point.

Graphs of and

𝒇𝒇 ′

Graphs of and (cont.)

𝒇 ′ (𝒙 )=𝟐 𝒙

Example 4Use the alternative form of the derivative.

Alternative Form of the Derivative

When is a function differentiable?

• Functions are not differentiable . . . • at sharp turns (v’s in the function),• when the tangent line is vertical, and• where a function is discontinuous.

Theorem 3.1 Differentiability Implies ContinuityIf is differentiable at , then is continuous at .

Example 5Describe the -values at which is differentiable.

a.

b.