1. 3.1 Derivative of a Function
is called the derivative of at .
We write:
“The derivative of f with respect to x is …”
There are many ways to write the derivative of

2. 3.1 Derivative of a Function
“f prime x” or
“the derivative of f with respect to x”
“y prime”
“dee why dee ecks”
“the derivative of y with respect to x”
“dee eff dee ecks”
“dee dee ecks uv eff uv ecks”
“the derivative of f of x”

3. 3.1 Derivative of a Function
Note:
dx does not mean d times x !
dy does not mean d times y !

4. 3.1 Derivative of a Function
Note:
does not mean !
(except when it is convenient to think of it as division.)
does not mean !
(except when it is convenient to think of it as division.)

5. 3.1 Derivative of a Function
Note:
does not mean times !
(except when it is convenient to treat it that way.)

6. 3.1 Derivative of a Function
The derivative is the slope of the original function.
The derivative is defined
at the end points of a
function on a closed
interval.

7. 3.1 Derivative of a Function

8. 3.1 Derivative of a Function
A function is differentiable if it has a derivative everywhere in its domain. It must be continuous and smooth. Functions on closed intervals must have one-sided derivatives defined at the end points.

9. 3.2 Differentiability
To be differentiable, a function must be continuous and smooth.
Derivatives will fail to exist at:
cusp
corner
vertical tangent
discontinuity

10. 3.2 Differentiability
Most of the functions we study in calculus will be differentiable.

11. 3.2 Differentiability There are two theorems on page 110:
If f has a derivative at x = a, then f is continuous at x = a.
Since a function must be continuous to have a derivative, if it has a derivative then it is continuous.

12. 3.2 Differentiability Intermediate Value Theorem for Derivatives
If a and b are any two points in an interval on which f is differentiable, then takes on every value between and
Between a and b, must take on every value between and .

13. 3.3 Rules for Differentiation
If the derivative of a function is its slope, then for a constant function, the derivative must be zero.
example:
The derivative of a constant is zero.

14. 3.3 Rules for Differentiation
We saw that if ,
This is part of a pattern.
examples:
power rule

15. 3.3 Rules for Differentiation
Proof:

16. 3.3 Rules for Differentiation
constant multiple rule:
examples:

17. 3.3 Rules for Differentiation
constant multiple rule:
sum and difference rules:
(Each term is treated separately)

18. 3.3 Rules for Differentiation
Find the horizontal tangents of:
Horizontal tangents occur when slope = zero.
Substituting the x values into the original equation, we get:
(The function is even, so we only get two horizontal tangents.)

19. 3.3 Rules for Differentiation

20. 3.3 Rules for Differentiation
First derivative (slope) is zero at:

21. 3.3 Rules for Differentiation
product rule:
Notice that this is not just the product of two derivatives.
This is sometimes memorized as:

22. 3.3 Rules for Differentiation
product rule:
add and subtract u(x+h)v(x)
in the denominator
Proof

23. 3.3 Rules for Differentiation
quotient rule: or

24. 3.3 Rules for Differentiation
Higher Order Derivatives:
is the first derivative of y with respect to x.
is the second derivative.
(y double prime)
is the third derivative.
We will learn later what these higher order derivatives are used for.
is the fourth derivative.

25. 3.3 Rules for Differentiation
Suppose u and v are functions that are differentiable at
x = 3, and that u(3) = 5, u’(3) = -7, v(3) = 1, and v’(3)= 4.
Find the following at x = 3 :

26. 3.3 Rules for Differentiation

27. 3.3 Rules for Differentiation

28. 3.4 Velocity and other Rates of Change
Consider a graph of displacement (distance traveled) vs. time.
Average velocity can be found by taking:
time (hours)
distance
(miles) B A
The speedometer in your car does not measure average velocity, but instantaneous velocity.
(The velocity at one moment in time.)

29. 3.4 Velocity and other Rates of Change
Velocity is the first derivative of position.
Acceleration is the second derivative
of position.

30. 3.4 Velocity and other Rates of Change
Gravitational
Constants:
Example:
Free Fall Equation
Speed is the absolute value of velocity.

31. 3.4 Velocity and other Rates of Change
Acceleration is the derivative of velocity.
example:
If distance is in:
Velocity would be in:
Acceleration would be in:

32. 3.4 Velocity and other Rates of Change
time
distance
acc neg
vel pos &
decreasing
acc neg
vel neg &
decreasing
acc zero
vel neg &
constant
acc zero
vel pos &
constant
acc pos
vel neg &
increasing
velocity
zero
acc pos
vel pos &
increasing
acc zero,
velocity zero

33. 3.4 Velocity and other Rates of Change
Average rate of change =
Instantaneous rate of change =
These definitions are true for any function.
( x does not have to represent time. )

34. 3.4 Velocity and other Rates of Change
For a circle:
Instantaneous rate of change of the area with
respect to the radius.
For tree ring growth, if the change in area is constant then dr must get smaller as r gets larger.

35. 3.4 Velocity and other Rates of Change
from Economics:
Marginal cost is the first derivative of the cost function, and represents an approximation of the cost of producing one more unit.

36. 3.4 Velocity and other Rates of Change
Example 13:
Suppose it costs:
to produce x stoves.
If you are currently producing 10 stoves, the 11th stove will cost approximately:
The actual cost is:
marginal cost
actual cost

37. 3.4 Velocity and other Rates of Change
Note that this is not a great approximation – Don’t let that bother you.
Marginal cost is a linear approximation of a curved function. For large values it gives a good approximation of the cost of producing the next item.

38. 3.4 Velocity and other Rates of Change

39. 3.5 Derivatives of Trigonometric Functions
Consider the function
slope
We could make a graph of the slope:
Now we connect the dots!
The resulting curve is a cosine curve.

40. 3.5 Derivatives of Trigonometric Functions
Proof

41. 3.5 Derivatives of Trigonometric Functions
= 0
= 1

42. 3.5 Derivatives of Trigonometric Functions
Find the derivative of cos x

43. 3.5 Derivatives of Trigonometric Functions
= 0
= 1

44. 3.5 Derivatives of Trigonometric Functions
We can find the derivative of tangent x by using the quotient rule.

45. 3.5 Derivatives of Trigonometric Functions
Derivatives of the remaining trig functions can be determined the same way.

46. 3.5 Derivatives of Trigonometric Functions
Jerk A sudden change in acceleration
Definition Jerk
Jerk is the derivative of acceleration. If a body’s position
at time t is s(t), the body’s jerk at time t is

47. 3.5 Derivatives of Trigonometric Functions

48. 3.6 Chain Rule
Consider a simple composite function:

49. 3.6 Chain Rule Chain Rule: If is the composite of and , then: Find:
example:
Find:

50. 3.6 Chain Rule

51. 3.6 Chain Rule Here is a faster way to find the derivative:
Differentiate the outside function...
…then the inside function

52. 3.6 Chain Rule The chain rule can be used more than once.
(That’s what makes the “chain” in the “chain rule”!)

53. 3.6 Chain Rule
Derivative formulas include the chain rule!
etcetera…

54. 3.6 Chain Rule
Find

55. 3.6 Chain Rule
The chain rule enables us to find the slope of parametrically defined curves:
The slope of a parametrized curve is given by:

56. 3.6 Chain Rule
Example:
These are the equations for an ellipse.

57. 3.7 Implicit Differentiation
This is not a function, but it would still be nice to be able to find the slope.
Do the same thing to both sides.
Note use of chain rule.

58. 3.7 Implicit Differentiation
This can’t be solved for y.
This technique is called implicit differentiation.
1 Differentiate both sides w.r.t. x.
2 Solve for

59. 3.7 Implicit Differentiation
Implicit Differentiation Process
Differentiate both sides of the equation with respect to x.
Collect the terms with dy/dx on one side of the equation.
Factor out dy/dx .
Solve for dy/dx .

60. 3.7 Implicit Differentiation
Find the equations of the lines tangent and normal to the curve at
Note product rule.

61. 3.7 Implicit Differentiation
Find the equations of the lines tangent and normal to the curve at
normal:
tangent:

62. 3.7 Implicit Differentiation

63. 3.7 Implicit Differentiation
Find if
Substitute
back into the equation.

64. 3.7 Implicit Differentiation
Rational Powers of Differentiable Functions
Power Rule for Rational Powers of x
If n is any rational number, then

65. 3.7 Implicit Differentiation
Proof: Let p and q be integers with q > 0.
Raise both sides to the q power
Differentiate with respect to x
Solve for dy/dx

66. 3.7 Implicit Differentiation
Substitute for y
Remove parenthesis
Subtract exponents

67. 3.8 Derivatives of Inverse Trigonometric Functions
Slopes are reciprocals.
Because x and y are reversed to find the reciprocal function, the following pattern always holds:
The derivative of
Derivative Formula for Inverses:
evaluated at
is equal to the reciprocal of
the derivative of
evaluated at

68. 3.8 Derivatives of Inverse Trigonometric Functions
We can use implicit differentiation to find:

69. 3.8 Derivatives of Inverse Trigonometric Functions
We can use implicit differentiation to find:
But
so is positive.

70. 3.8 Derivatives of Inverse Trigonometric Functions

71. 3.8 Derivatives of Inverse Trigonometric Functions
Find

72. 3.8 Derivatives of Inverse Trigonometric Functions
Find

73. 3.8 Derivatives of Inverse Trigonometric Functions

74. 3.8 Derivatives of Inverse Trigonometric Functions
Your calculator contains all
six inverse trig functions.
However it is occasionally
still useful to know the following:

75. 3.8 Derivatives of Inverse Trigonometric Functions
Find

76. 3.9 Derivatives of Exponential and Logarithmic Functions
Look at the graph of
If we assume this to be true, then:
The slope at x = 0 appears to be 1.
definition of derivative

77. 3.9 Derivatives of Exponential and Logarithmic Functions
Now we attempt to find a general formula for the derivative of using the definition.
This is the slope at x = 0, which we have assumed to be 1.

79. 3.9 Derivatives of Exponential and Logarithmic Functions
is its own derivative!
If we incorporate the chain rule:
We can now use this formula to find the derivative of

80. 3.9 Derivatives of Exponential and Logarithmic Functions
Incorporating the chain rule:

81. 3.9 Derivatives of Exponential and Logarithmic Functions
So far today we have:
Now it is relatively easy to find the derivative of

82. 3.9 Derivatives of Exponential and Logarithmic Functions

83. 3.9 Derivatives of Exponential and Logarithmic Functions
To find the derivative of a common log function, you could just use the change of base rule for logs:
The formula for the derivative of a log of any base other than e is:

84. 3.9 Derivatives of Exponential and Logarithmic Functions

85. 3.9 Derivatives of Exponential and Logarithmic Functions
Find y’

86. 3.9 Derivatives of Exponential and Logarithmic Functions
Logarithmic differentiation
Used when the variable is in the base and the exponent
y = xx
ln y = ln xx
ln y = x ln x