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 x0 is not a critical point, f ‘(x0)≠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
http://webspace.ship.edu/msrenault/GeoGebraCalculus/derivative_as_a_function.html
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 y = f '(x) or y = dy
dx
Animations for Some Derivative
Functions
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Linear function (horizontal)
Linear function (inclined)
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)=x2, 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.
1
a. y = x d. y = x e. y = 2
x
4
3
Basic Derivatives