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The Derivative

- The Definition of the DerivativeSlope of the

Tangent LineDerivative and DifferentiabilityDeri

vatives GraphicallyDifferentiability and

ContinuityDerivatives as Functions

The Definition of the Derivative

Definition

The Definition of the Derivative

Definition

Notations

Recall that by previous considerations, the

derivative of a function f at the point x0 is

the slope of the tangent line of the graph of f

at the point (x0,f(x0)). This is an important

geometric view to the derivative.

Slope of the Tangent Line (1)

To compute the equation of the tangent line, at a

point (a,f(a)) of the graph of a function f

observe that

Slope of the Tangent Line (2)

Example

Slope of the Tangent Line (2)

Example

Solution

Slope of the Tangent Line (2)

Example

Solution

Slope of the Tangent Line (2)

Example

Solution

Slope of the Tangent Line (2)

Example

Solution

Slope of the Tangent Line (3)

Example

Solution (contd)

Conclusion

Derivative and Differentiability (1)

Definition 1

Derivative and Differentiability (1)

Definition 1

Definition 2

Derivative and Differentiability (1)

Definition 1

Definition 2

Derivative and Differentiability (2)

Theorem

Derivative and Differentiability (2)

Theorem

Proof

?

Derivative and Differentiability (3)

Theorem

Proof (contd)

?

Differentiability Graphically

Differentiability Graphically

Differentiability Graphically

Differentiability Graphically

Differentiability and Continuity

Theorem

Proof

The Derivative as a Function

Definition

Notation

The derivative of a function f is a function

whose the domain of definition consists of all

the numbers x such that the function f has

its derivative at the point x.

Graphs of Functions and their Derivatives

Problem

The picture on the right shows the graphs of two

functions f and g and the graphs of their

derivatives. The maximum of the shown values of

f is larger than that of g. Which is which?

Graphs of Functions and their Derivatives

Df

Problem

f

The picture on the right shows the graphs of two

functions f and g and the graphs of their

derivatives. The maximum of the shown values of

f is larger than that of g. Which is which?

Solution

The red curve appears to intersect the x-axis

whenever the tangent of the blue curve is

horizontal. Hence the red curve is the graph of

the derivative of the function f. The blue

curve is the graph of f.

g

Dg

The green curve is everywhere decreasing. Hence

its tangents have negative slopes. The black

curve is the only curve that takes only negative

values. Hence the green curve is the graph of

the function g and the black curve that of its

derivative.

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