Derivative of a Function Ch 3.1 Derivative of a Function

What is a derivative? When m = lim ℎ→0 𝑓 𝑎+ℎ −𝑓(𝑎) ℎ exists, the limit is called the derivative of f at a. We write f’(x) = lim ℎ→0 𝑓 𝑥+ℎ −𝑓(𝑥) ℎ The set of points in the domain of f’ for which the limit exists may be smaller than the domain of f. A function that is differentiable at every point of its domain is a differentiable function.

EX 1: Applying the definition of the Derivative Function: f(x) = x³

What is the “alternate definition” of a derivative at a point What is the “alternate definition” of a derivative at a point? Relabel the same graph: Therefore the derivative of a function f at the point x=a is the limit: 𝐥𝐢𝐦 𝒙→𝒂 𝒇 𝒙 −𝒇(𝒂) 𝒙−𝒂

EX 2: Applying the alternative definition of the derivative Differentiate f(x) = 𝑥 3 using the alternate definition of the derivative ( lim 𝑥→𝑎 𝑓 𝑥 −𝑓(𝑎) 𝑥−𝑎 )

You Try! Differentiate f(x) = 𝑥 using the alternate definition of the derivative ( lim 𝑥→𝑎 𝑓 𝑥 −𝑓(𝑎) 𝑥−𝑎 )

NOTE THE PATTERNS and remember the Power Rule  Function: f(x) = 𝑥 3 f(x) = 𝑥 = 𝑥 1/2 Derivative: f’(x) = 3x² f’(x) = 1 2 𝑥 = 1 2 𝑥 −1/2 on your hw I have instructions on which problems to do with the power rule and which ones you must do with the definitions.

What notation should I be familiar with? There are many ways to denote the derivative of a function y = f(x). These are the most common: 𝑑𝑓 𝑑𝑥 Notation How to say it Comments f’(x) “f prime of x” y’ “y prime” Nice and brief, but does not name independent variable 𝑑𝑦 𝑑𝑥 “dy dx” or “the derivative of y with respect to x” Names both variables and uses d for derivative 𝑑𝑓 𝑑𝑥 “df dx” or “the derivative of f with respect to x” Emphasizes the functions name 𝑑 𝑑𝑥 𝑓(𝑥) “d dx of f at x” or “the derivative of f at x” Emphasizes the idea that differentiation is an operation performed on f

What is the relationship between the graph 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. Ex 3A: Graphing f’ from f f’(x) = f(x)=x² Graph the derivative of the function f whose graph is shown in the figure. Note how the odd functions derivative is even!!!

If the slope of f is negative, the graph of f’ will be below the x-axis

Ex 3B: Graphing f’ from f Graph the derivative of the function f whose graph is shown in the figure. Note how the odd functions derivative is even!!!

You Try! Match each graph f(x) on the left with the graph of its derivative below.

EX 4: Graphing f from f’ Sketch the graph of a function f that has the following properties: f(0) = 0 The graph of f’ is as shown in the following figure: f is continuous for all x

EX 6: One-Sided Derivatives can Differ at a Point Show that the function f(x)= 𝑥 2 , 𝑥≤0 2𝑥, 𝑥>0 has left-hand and right-hand derivatives at x = 0, but there is no derivative there. *First check for continuity, then check derivatives

You Try! Using one-sided derivatives, show that the following function does not have a derivative at x = 2 F(x) = 𝑥 2 +𝑥+1, 𝑥≤2 4𝑥−3, 𝑥>2

Ch 3 Assignment 3.1: pg 105 QR #1-10all EX #1, 2 (use def), 3 and 4 (use power rule), 5-7 (use def), 18-20 (use power rule), 21, 26, 31, 32