3.1 Derivatives of a Function, p. 98 AP Calculus AB/BC.

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3.1 Derivative of a Function

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3.1 Derivatives of a Function, p. 98 AP Calculus AB/BC

This limit is also called the derivative of f at a. We write: “The derivative of f with respect to x is …” In section 2.4, the slope of the curve of y = f(x) was defined at the point where x = a to be

Example 1: 0

Definition (Alternate) Derivative at a Point The derivative of a function at a point x = a is: provided that the limit exists.

Example 2: Use the alternate definition to differentiate For a = 2: −1−1 There are many ways to write the derivative of

“ f prime x ” or “the derivative of f with respect to x ” “ y prime” “the derivative of y with respect to x ” “the derivative of f with respect to x ” “the derivative of f of x ”

dx does not mean d times x ! dy does not mean d times y !

does not mean ! (except when it is convenient to think of it as division.) does not mean ! (except when it is convenient to think of it as division.)

(except when it is convenient to treat it that way.) does not mean times !

The derivative is the slope of the original function. The derivative is defined at the end points of a function on a closed interval. Example 3: Graph the derivative of the function f.

Graph y and y′ where

A function is differentiable if it has a derivative everywhere in its domain. It must be continuous and smooth. Functions on closed intervals must have one-sided derivatives defined at the end points. 