Derivative of a Function Chapter 3 Derivatives Section 3.1 Derivative of a Function

Bellwork

Quick Review Solutions

Quick Review

Quick Review Solutions

What you’ll learn about The meaning of differentiable Different ways of denoting the derivative of a function Graphing y = f (x) given the graph of y = f ′(x) Graphing y = f ′(x) given the graph of y = f (x) One-sided derivatives Graphing the derivative from data … and why The derivative gives the value of the slope of the tangent line to a curve at a point.

Definition of Derivative

Differentiable Function

Example Definition of Derivative

Derivative at a Point (alternate)

Notation

Relationships between the Graphs of f and f ′ Because we can think of the derivative at a point in graphical terms as slope, 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. We estimate the slope of the graph of f in y-units per x-unit at frequent intervals. We then plot the estimates in a coordinate plane with the horizontal axis in x-units and the vertical axis in slope units.

Graphing the Derivative from Data Discrete points plotted from sets of data do not yield a continuous curve, but we have seen that the shape and pattern of the graphed points (called a scatter plot) can be meaningful nonetheless. It is often possible to fit a curve to the points using regression techniques. If the fit is good, we could use the curve to get a graph of the derivative visually. However, it is also possible to get a scatter plot of the derivative numerically, directly from the data, by computing the slopes between successive points.

One-sided Derivatives

One-sided Derivatives Right-hand and left-hand derivatives may be defined at any point of a function’s domain. The usual relationship between one-sided and two-sided limits holds for derivatives. Theorem 3, Section 2.1, allows us to conclude that a function has a (two-sided) derivative at a point if and only if the function’s right-hand and left-hand derivatives are defined and equal at that point.

Example One-sided Derivatives

Homework Exercise 1-35 odds