3.1 Derivative of a Function Definition Alternate Definition One-sided derivatives Data Problem

Derivative? Differential calculus is the study of the rate of change of a function. The DERIVATIVE of the function f at x = a is defined to be: provided the limit exists. In the last chapter we defined this as the slope of the curve f at x = a. Now we have a specific name for this limit and will expand the use to include a broader spectrum of applications.

The DERIVATIVE of f with respect to x (general derivative) is defined in a similar manner (notice the notation!): Since this yields an expression in terms of x, the DERIVATIVE IS A FUNCTION! The domain of the derivative may be equal to the domain of f or less than the domain of f, never greater than the domain of f.

Differentiable Function If the derivative, f ` (x), exists we say that the function f (x) is differentiable at x. A function differentiable at every point in its domain is called a differentiable function. Differentiate the functions:

Alternate Definition of the Derivative An easier version of the derivative to be used only when seeking a value at a specific point: provided the limit exists.

Differentiate using the Alternate Definition of the Derivative: This implies that the general derivative is:

One-sided Derivatives: A function y = f(x) is differentiable on a closed interval [ a, b ] if it has a derivative at every point in the interior of the interval, and if the limits exist at the endpoints. Right-hand derivative Left-hand derivative

Let’s look at the following piecewise function, (this is mostly where we use one-sided derivatives): We want to examine the left, right, and undirected derivatives at the point where the function is split in its definition, x = 1.

Notation The first three notations are used to describe the result of taking the derivative, also lets us call the derivative something. The last three are strictly to indicate that an operation (taking the derivative) needs to be performed on the function called y or f. Function names need not be f or y, they can be g, h or any other variable you’d like to use.

y prime, brief form of the notation but does not name the independent variable. The derivative of y with respect to x or dy dx, this names both variables and uses d for derivative. df dx or the derivative of f with respect to x, emphasizes the name of the function. The derivative of at x, emphasizes the idea that differentiation is an operation performed on f.

Graph f and f ` Let’s graph the function and its derivative. Estimate the slope around x = -1 Estimate the slope around x = -1.5 Estimate the slope around x = 0 Estimate the slope around x = 1 Begin by graphing the function on the interval [ -2, 2 ]. Estimate the slope around x = 1.5

Graphing the function and its derivative:

Now we’re going to turn this around, I want you to sketch a graph of a function such that f (0) = 0, f is continuous for all x, and the graph of the derivative is the following graph. Does the derivative exist at x = 1? What are the one-sided derivative values at x = 1?