3.1 Derivatives of a Function

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3.1 Derivative of a Function

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Presentation transcript:

3.1 Derivatives of a Function AP Calculus AB 3.1 Derivatives of a Function

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

Example 1:

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 −1 For a = 2: There are many ways to write the derivative of

“the derivative of f with respect to x” “f prime x” or “y prime” “the derivative of y with respect to x” “dee why dee ecks” or “the derivative of f with respect to x” “dee eff dee ecks” or “the derivative of f of x” “dee dee ecks uv eff uv ecks” or

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

Note: does not mean ! 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.)

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

In the future, all will become clear.

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

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. p