The Derivative Eric Hoffman Calculus PLHS Oct. 2007

Key Topics Derivative: the rate of change of a function f(x) at any number x for which it is defined –Recall: rate of change really means slope –In section 2.2 we learned how to calculate the slope of the tangent line at a point (a,b) The derivative of a function is another function which is denoted by f ` (f prime) The derivative function gives you the slope of the function f at any point along f(x)derivative

Key Topics By using our knowledge of calculating the slope of the line tangent to f(x) we can generalize this for all x = x 0 and we can define the derivative as:

Key Topics Ex. Calculate the derivative of the function f(x) = x 2

Key Topics Calculate the derivative of the following functions: Remember:

Key Topics Other notations for derivatives: Note: All of the above expressions denote the derivative of y = f(x) and are read as “the derivative of ___ with respect to x” or “the derivative with respect to x”

Rules for Calculating Derivatives Rule 1: if f is a constant function f(x) = c thenRule 1 f `(x) = 0 Rule 2: if f(x) = mx + b then f `(x) = mRule 2 Rule 3 (power rule): let n ε R with n ≠ 0.Rule 3 If f(x) = x n, then f `(x) = nx n-1

Properties of Derivatives 1.If the function h = f + g is differentiable, then 2.If the function r = cf, where c is a constant, is differentiable, then: This means f `(x) exists for all x

Key Topics Homework pg all I will pick a few problems from section 2.3 and 2.4 that will be due on next Monday. Word of advice, ask questions over any problem that you don’t understand, any problem that is assigned is fair game for being part of the assigment

Property 1 Find the following:

Property 2 Find the following:

Rule 1 Find the following:

Rule 2 Find the follwing:

Rule 3 Find the following: