3.1 Derivative of a Function
Why?
-To learn how to take derivatives and find slopes
-To learn how to graph derivatives from functions and vice versa.
-Because derivatives allow you to find instantaneous rates of change and have many real life applications!
Definition of a Derivative:
The derivative of a function f with respect to the variable x is the function f’ whose value at x is
𝑓′(𝑥) = lim
ℎ→0
𝑓(𝑥+ℎ)−𝑓(𝑥)
ℎ
provided the limit exists.
If a function has a derivative (the limit exists) at every point in its domain, then it is a differentiable function.
Ex. 1 Find the derivative of y = x3
Alternate Definition:
The derivative of the function f at the point x = a is the limit
𝑓′(𝑥) = lim
𝑥→𝑎
𝑓(𝑥)−𝑓(𝑎)
𝑥−𝑎
provided the limit exists.
Ex. 2 Differentiate 𝑓(𝑥) = √𝑥
Notation:
How it’s read
y’
𝑑𝑦
𝑑𝑦
𝑑𝑓
𝑑𝑥
𝑑
𝑓(𝑥)
𝑑𝑥
Ex. 3 Graphing f’ from f
Ex. 4 Graphing f from f’
“y prime”
“dy dx”
“the derivative of y with respect to x”
“df dx”
“the derivative of f with respect to x”
“d dx of f”
“the derivative of f at x”
Why use it?
quick
uses both variables
Uses the name of the function
Emphasizes that differentiation is an
operation performed on f
One-Sided Derivatives
The derivative can be different on either side of a point. If the right-hand derivative is not equal to the left-hand
derivative, then the derivative does not exist at that point.
Right side: 𝑓′(𝑥) = lim
ℎ→0+
Left side: 𝑓′(𝑥) = lim
ℎ→0−
𝑓(𝑥+ℎ)−𝑓(𝑥)
ℎ
𝑓(𝑥+ℎ)−𝑓(𝑥)
ℎ
Ex. 5 Find the left and right side derivatives at x=0. Is there a derivative at x=0?
𝑦 = 𝑓(𝑥) = {
𝑥2,
2𝑥,
𝑥≤0
𝑥>0
3.2 Differentiability
Why?
-To learn when derivatives exist
-Because derivatives allow you to find instantaneous rates of change and have many real life applications!
How f’(a) Might Fail to Exist
In 3.1, we said that the derivative exists as long as the right-hand derivative is equal to the left-hand derivative.
ℎ→0+
lim
𝑓(𝑥+ℎ)−𝑓(𝑥)
ℎ
lim
𝑓(𝑥)−𝑓(𝑎)
𝑥−𝑎
𝑓(𝑥+ℎ)−𝑓(𝑥)
ℎ
ℎ→0−
= lim
or alternately if
𝑥→𝑎
exists
There are four instances that we see on graphs where the derivative fails to exist.
1. a corner
2. a cusp
3. a vertical tangent
4. a discontinuity
Differentiability Implies Local Linearity
Differentiable functions are locally linear. This means that if you zoom in enough on any point, it will eventually look like
a (straight) line. If the function is not differentiable at a point, then you will not see the function “straighten out.”
Exploration: For the two functions below… (do separately)
1) Graph the function in the standard window.
2) Is there a point where the functions appear to not be differentiable? If so, where?
3) Zoom in on that point several times.
4) What happens when you continue to zoom in? Does it change your answer to (2)?
g(x) = √𝑥 2 + 0.0001 + .99
f(x) = |x| + 1
2)
2)
4)
4)
5) Graph f(x) and g(x) at the same time in the standard window. What do you see?
6) Zoom in on the point (0,1). What do you see?
𝟐⁄
𝟑.
7) Graph 𝒚 = 𝒙𝟑 + 𝟎. 𝟏(𝒙 − 𝟏)
point?
Is the function differentiable at (1,1)? What happens when you zoom in on the
Derivatives on the calculator
Ex. 1 Use the calculator to find the value of the derivative of y=x3 at x=2.
Ex. 2 Use the calculator to graph the derivative of y=x3.
Can we always trust a calculator?
Ex. 3 Use the calculator to find the value of f’(0) for f(x) = |x|
Differentiability Implies Continuity
If f has a derivative at x = a, then f is continuous at x = a.
3.3 Rules for Differentiation
Why?
-To learn how to take derivatives with powers, products, and quotients.
-Because derivatives allow you to find instantaneous rates of change and have many real life applications!
Rule 1: Derivative of a Constant function
If f is the function with the constant value c, then
Rule 2: Power Rule for positive and negative powers of x
If n is a positive integer, then
Rule 3: The Constant Multiple Rule
If u is a differentiable function of x and c is a constant, then
Rule 4: The Sum and Difference Rule
If u and v are differentiable functions of x, then their sum and difference are differentiable at every point where u and v
are differentiable. At such points,
Rule 5: The Product Rule
The product or two differentiable functions u and v is differentiable, and
Rule 6: The Quotient Rule
At a point where v≠0, the quotient y = u/v of two differentiable functions is differentiable, and
Ex. 1 Does the curve y = x4-2x2+2 have any horizontal tangents? If so, where?
Ex. 2 Find the derivative of 𝑓(𝑥) =
𝑥2
𝑥
3 different ways.
Ex. 3 Let u and v be functions differentiable at x = 3 such that u(3) = 1, u’(3) = 2, v(3) = -2, and v’(3)=-4, find the value of
the following derivatives at x = 3
𝑑
𝑑
a) 𝑑𝑥 (𝑢𝑣)
𝑢
𝑑
b) 𝑑𝑥 (𝑣 )
c) 𝑑𝑥 (2𝑢 − 3𝑣)
Second and Higher Order Derivatives
𝑑𝑦
y' = 𝑑𝑥 is called the first derivative of y. If it is differentiable, then we can take the derivative again to get the second
𝑑2 𝑦
𝑑3 𝑦
𝑑𝑛𝑦
derivative y’’ or 𝑑𝑥 2 . The third derivative can be written y’’’ or 𝑑𝑥 3 . After that, we write y(n) or 𝑑𝑥 𝑛 .
Ex. 4 Find the first 4 derivatives of the function.
a) y = 3x2 - 4x + 2
b) y = x2 + x-2
3.4 Velocity and Other Rates of Change
Why?
-To learn how to find velocity and acceleration
-Because derivatives allow you to find instantaneous rates of change and have many real life applications!
Instantaneous Rate of Change
The instantaneous rate of change of f with respect to x at a is the derivative
Displacement
The displacement of the object over the time interval from t to t+∆t is
Average Velocity
The average velocity from t to t+∆t is
Instantaneous Velocity
The velocity is the derivative of the position function s = f(t) with respect to time. At time t the velocity is
Speed
Speed is the absolute value of velocity
Acceleration
Acceleration is the derivative of velocity with respect to time. If a body’s velocity at time t is v(t) = ds/dt, then the body’s
acceleration at time t is
Ex. 1 A dynamite blast propels a heavy rock straight up with a launch velocity of 160 ft/sec. It reaches a height of
s = 160t2 -16t2 after t seconds.
a) How high did the rock go?
b) What is the velocity and speed of the rock when it is 256 ft. above the ground on the way up? On the way down?
c) What is the acceleration of the rock at any time t during its flight?
d) When did the rock hit the ground?
Ex. 2 A particle moves along a line so that its position at any time t ≥ 0 is given by the function x(t) = t2 -4t + 3, where x is
measured in meters and t is measured in seconds.
a) Find the displacement of the particle in the first 2 seconds.
b) Find the average velocity of the particle during the first 4 seconds.
c) Find the velocity of the particle when t = 4.
d) Find the acceleration of the particle when t = 4.
e) Describe the motion of the particle. At what values of t does the particle change directions?
3.5 Derivatives of Trigonometric Functions
Why?
-To learn how to take derivatives of the 6 basic trigonometric functions.
-Because derivatives allow you to find instantaneous rates of change and have many real life applications!
Exploration
1) Use your calculator to graph the derivative of y = sinx. What do you think the derivative of sinx is?
2) Graph your guess in the same window as the derivative. Is your guess correct? If not, correct it.
3) Use your calculator to graph the derivative of y = cosx. What do you think the derivative of cosx is?
4) Graph your guess in the same window as the derivative. Is your guess correct? If not, correct it.
𝑑
(𝑠𝑖𝑛𝑥) =
𝑑𝑥
𝑑
(𝑐𝑜𝑠𝑥) =
𝑑𝑥
Ex. 1 Find the derivative of
a) y = x2cosx
b) 𝑓(𝑥) =
𝑐𝑜𝑠𝑥
1−𝑠𝑖𝑛𝑥
Simple Harmonic Motion
The motion of a weight bobbing up and down on a spring is an example of simple harmonic motion. (Our examples
ignore any other forces, like friction.)
Ex. 2 A weight hanging from a spring is stretched 5 units beyond its rest position ( s = 0 ) and released at time t=0 to bob
up and down. Its position at any later time t is s=5cost. What are the velocity and acceleration at time t? Describe its
motion.
Derivatives of the Other Basic Trigonometric Functions
Ex. 3 Use the quotient rule to find the derivative of y = tanx
𝑑
(𝑡𝑎𝑛𝑥)
𝑑𝑥
=
𝑑
(𝑐𝑜𝑡𝑥)
𝑑𝑥
=
𝑑
(𝑠𝑒𝑐𝑥)
𝑑𝑥
=
𝑑
(𝑐𝑠𝑐𝑥)
𝑑𝑥
=
Ex. 4 Find an equation for the line that is normal to 𝑦 =
Ex. 5 Find y’’ of y = csc x
𝑠𝑒𝑐𝑥
𝑥
at x = 2