The Derivative Function

Warming UP Exercise 7 from Derivative at a Point Consider the graph below. The domain of the function is all the real numbers. Assume that outside the window the function continues the same behavior as the one indicated in the window. 1.Where is f(x) increasing? 1.Where is f(x)>0? 1.Where is f(x) concave up?

i. Sketch the tangent line at each of the given points and use the grid to complete the table below. All the answers are estimates

ii. Use interval notation to complete the following information a. Intervals where the derivative is negative (solutions to f ‘ (x)<0) b. Intervals where f(x)<0. Describe those points graphically. c. Intervals where the derivative is positive (solutions to f ‘ (x)>0) d. Intervals where f (x)>0. Describe those points graphically.

Critical Points of a Continuous Function A critical points of a continuous function y=f(x) is a point in its domain where f ‘ (x)=0 or f ‘(x) is undefined. f’(x)=0 when the tangent line is horizontal f’(x) is undefined at a point in the domain where the tangent line does not exist (cusp, corner, end point), or when the tangent line is vertical.. If x 0 is not a critical point, f ‘(x 0 )≠0

Exercise 1 The first coordinate of the critical points of each of the functions below are identified at the top of each graph. Refer to the definition of a critical point to explain why it is a critical point. Identify the type of critical point (f’=0 or f’ undefined)

Exercise

Questions Identify all the critical points on the given domain Determine the sign of the derivative between any two critical points Estimate the derivative (draw tangent lines to find them) at x=-2, 0, 2, 4, 6 Compare results with the applet Analyze the graph of f ‘(x)

Derivative Function Now a new function is defined in such a way that to each point in the domain of the function y = f(x) is assigned the value of the derivative at that point, or what is the same the value of the slope of the tangent line. This new function is called the derivative function of y = f(x). The derivative function of y=f(x) is denoted

Animations for Some Derivative Functions Linear function (horizontal) Linear function Linear function (inclined) Linear function Quadratic function Sine function Y=|x| A non-differentiable function

Deriving Basic Derivative Formulas

If f(x)=c, constant f ‘( c )=0

y=m x + b, y ‘=m

y(x)=x 2, y ‘(x)=2x

Derivative of a Power Function

Exercise 5 Rewrite each of the following functions as a power function. Use the shortcut for the derivative of power functions to find the derivative. Give the final answer with positive exponents.

Basic Derivatives