 The derivative of a function f(x), denoted f’(x) is the slope of a tangent line to a curve at any given point.  Or the slope of a curve at any given point  Or the instantaneous rate of change of y with respect to x.

What is a tangent line?

 Find the slope of the tangent line to the graph of f at the given x value 1. f(x) = x 2 at x = 2 2. f(x) = x 2 + 2x at x = -2  Answer to 1) f’(2) = 4  Answer to 2) f’(-2) = -2

 The derivative function for example 1 is…  f’(x) = 2x  f’(2) = 4  The derivative function for example 2 is…  f’(x) = 2x + 2  f’(-2) = -2

Rules to find functions that will give you the slope of the tangent lines to a curve at any value of x

 The derivative of any constant is 0.  That is….

 For any real number ‘n’

Find the derivative of each function below.

 Find the equation of the tangent line to the graph of f at the specified value of x. f(x) = 3x 4 – 2x + 1 at x = 0  Find the value(s) of “x” where the function has a horizontal tangent line

Find the derivative of each function below.

 Derivatives of sine and cosine

 Derivatives of natural logarithm and e x

 DERIVATIVES OF DERIVATIVES  THE RATE OF CHANGE OF A RATE OF CHANGE  NOTATION:  If y or f(x)  1 st derivative y’ or f’(x)  2 nd derivative y’’ or f”(x)  3 rd derivative y’’’ or f’’’(x)  4 th or more y (4) or f (4) (x)

 Find the 1 st and 2 nd derivatives of each.

The position of a car is given by the values in the following table. t(sec) s(feet) a) Graph the position function on graph paper b) Use your graph to find the average velocity for the time periods below i) 2 to 5 seconds ii) 2 to 3 seconds c) Estimate the instantaneous velocity when t = 2

 An average rate of change of a function is ALWAYS found by finding the slope of a secant line from time “a” to time “b”.  An Instantaneous rate of change of a function is found by finding the slope of the tangent line at time “a”. (the derivative!)  Sometimes an instantaneous rate can be estimated with an average rate if an equation is not known!

A ball is dropped from the top of a tower 450 meters above the ground. Its distance after ‘t’ seconds is given by the equation Find: a) The average velocity from t = 1 to t = 3 seconds b) The instantaneous velocity at t = 5 seconds.

a) b)

 A particle moves along a line such that after ‘t’ seconds its position is given by the equation a) Find the velocity of the particle at exactly 2 seconds. b) Find the acceleration at 1 second.

 Suppose that f and g are differentiable functions. Then….

 Suppose that f and g are differentiable functions and g(x)  0. Then….

 If y = sin x y’ = cos x  If y = cos x y’ = -sin x  If y = tan x y’ = sec 2 x  If y = cot xy’ = -csc 2 x  If y = sec xy’ = sec x tan x  If y = csc xy’ = -csc x cot x

 Rule for taking the derivative of functions that are a composition of two or more functions.  y= f(g(x))

 Find the derivative of each function.