3.1 Derivative of a Function

ObjectivesStudents will be able to:

1) Calculate slopes and derivatives using the definition of the derivative

2) Graph f’ from f, graph f from f’, and graph the derivative of a function given

numerically with data

Recall:

Alternate Definition of Derivative at a Point

• If f’(x) exists at a value x, then f(x) is differentiable at that value.

• If f’(x) exists for all x in the domain of f(x) then f(x) is a differentiable function.

Relationships between the Graphs of f and f’

• We can think of the derivative at a point in graphical terms as slope.

• Therefore, we can get a good idea of what the graph of the function f’ looks like by estimating the slopes at various points along the graph of f.

• The slope at a given x value of f will be the y-coordinate of the same x value on f’.

Ex 3: Draw a sketch of the derivative of the function f.

Try this one.

One-Sided Derivatives• A function y=f(x) is differentiable on a closed

interval [a,b] if it has a derivative at every interior point of the interval, and if the limits (seen below) exist at the endpoints

Ex 4: Using one-sided derivatives, show that the function does not have a derivative at x=0.