is called the derivative of at .The derivative of f with respect to x is 3.1 Derivative of a Function

3.1 Derivative of a Function

dx does not mean d times x !dy does not mean d times y !3.1 Derivative of a Function

(except when it is convenient to think of it as division.)(except when it is convenient to think of it as division.)3.1 Derivative of a Function

(except when it is convenient to treat it that way.)3.1 Derivative of a Function

The derivative is the slope of the original function.The derivative is defined at the end points of a function on a closed interval.3.1 Derivative of a Function

3.1 Derivative of a Function

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.3.1 Derivative of a Function

To be differentiable, a function must be continuous and smooth.Derivatives will fail to exist at:3.2 Differentiability

Most of the functions we study in calculus will be differentiable.3.2 Differentiability

There are two theorems :Since a function must be continuous to have a derivative, if it has a derivative then it is continuous.3.2 Differentiability

Intermediate Value Theorem for Derivatives3.2 Differentiability

If the derivative of a function is its slope, then for a constant function, the derivative must be zero.example:3.3 Rules for Differentiation

This is part of a pattern.examples:3.3 Rules for Differentiation

3.3 Rules for DifferentiationProof:

examples:constant multiple rule:3.3 Rules for Differentiation

(Each term is treated separately)constant multiple rule:sum and difference rules:3.3 Rules for Differentiation

Find the horizontal tangents of: Substituting the x values into the original equation, we get:

(The function is even, so we only get two horizontal tangents.)3.3 Rules for Differentiation

3.3 Rules for Differentiation

3.3 Rules for Differentiation

product rule:Notice that this is not just the product of two derivatives.This is sometimes memorized as:3.3 Rules for Differentiation

product rule:3.3 Rules for DifferentiationProofadd and subtract u(x+h)v(x)in the denominator

quotient rule:or3.3 Rules for Differentiation

Higher Order Derivatives:(y double prime)We will learn later what these higher order derivatives are used for.3.3 Rules for Differentiation

3.3 Rules for DifferentiationSuppose u and v are functions that are differentiable atx = 3, and that u(3) = 5, u(3) = -7, v(3) = 1, and v(3)= 4.Find the following at x = 3 :

3.3 Rules for Differentiation

3.3 Rules for Differentiation

***************************