Warm-Up: October 2, 2017 Find the slope of at

Derivative of a Function Section 3.1

Chapter 3 Overview We learned how to find the slope of a tangent to a curve as the limit of the slopes of secant lines. This gives us the instantaneous rate of change at a point. The study of rates of change of functions is called differential calculus.

Derivative The definition of derivative is the same as the function for slope. Provided the limit exists!!

Derivative Notation

Derivative Vocabulary If f’(x) exists, we say that f is differentiable at x. This is the same as saying “f has a derivative at x” A function that is differentiable at every point of its domain is a differentiable function. A function must be continuous in order to be differentiable.

Example Differentiate 𝑓 𝑥 = 𝑥 3

Definition (Alternate) The derivative of the function at a point a, is the limit 𝑓 ′ 𝑎 = lim 𝑥→𝑎 𝑓 𝑥 −𝑓 𝑎 𝑥−𝑎 Provided the limit exists.

Example Differentiate 𝑥 .

Warm-Up: October 3, 2017 Find the average rate of change of y=3x2 over the interval [-1, 4].

Graphing f’ from f NOTE: The domain of f’ may be smaller than the domain of f. Calculate/estimate the slope for various values of x. Plot the (x, y’) points Connect the points with a smooth curve for x-values where f is smooth and continuous.

Example 1

Graphing f from f’ You must be given at least one point. Start your graph at the given point.

Example 2 Sketch the graph that has the following properties 𝑓 0 =0 The graph of 𝑓′ is as shown in the figure. 𝑓 is continuous for all x.

One-Sided Derivatives Right-Hand Derivative Left-Hand Derivative Derivatives at endpoints are one-sided limits.

Example 3 𝑦= 𝑥 2 +1, 𝑥<2 −𝑥+3, 𝑥≥2

Derivatives from Data Open you textbook to page 103 Look at #20