1. The derivative as the slope of the tangent line
(at a point)

2. Video help:
MIT!!!

3. A function, which gives the:
What is a derivative?
A function, which gives the:
the rate of change of a function in general
the slope of the line tangent to the curve in general

4. What is a differential quotient?
Just a number!
the rate of change of a function at a given point
the slope of the line tangent to the curve at a certain point
The substitutional value of the derivative

5. The tangent line
single point
of intersection

6. slope of a secant line
f(a) - f(x)
a - x
f(x)
f(a) x a

7. slope of a (closer) secant line
f(a) - f(x)
a - x
f(x)
f(a) x x a

8. closer and closer… a

9. watch the slope...

10. watch what x does... x a

11. The slope of the secant line gets closer and closer to the slope of the tangent line...

12. As the values of x get closer and closer to a!

13. The slope of the secant lines gets closer
to the slope of the tangent line...
...as the values of x
get closer to a
Translates to….

14. Differential quotient
f(x) - f(a)
lim
x - a x a
as x goes to a
Equation for the slope
Which gives us the the exact slope
of the line tangent to the curve at a!

15. Differential quotient: other form
f(x+h) - f(x)
lim
h0
= f(x+h) - f(x)
(x+h) - x h
f(a+h) h
f(a)
a+h a
(For this particular curve, h is a negative value)

16. Velocity and other Rates of Change
Average rate of change =
Instantaneous rate of change =
These definitions are true for any function.

17. Velocity and other Rates of Change- physical menaing of the differential quotient
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.)

18. Velocity and other rates of change
Velocity is the first derivative of position.
Acceleration is the second derivative
of position.

19. Velocity Free Fall Equation Speed is the absolute value of velocity.
Gravitational
Constants:
Example:
Free Fall Equation
Speed is the absolute value of velocity.

20. 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:

21. 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
acc pos
vel pos &
increasing
velocity
zero
acc zero,
velocity zero

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

23. Theorem : f is differentiable on the interval (a,b)
Theorem : f is differentiable on the interval (a,b).  f is continuous on the interval (a,b).
Proof: Assume that f ’(c) exists for any c in (a,b).
f ’(c) = lim f(c+h) -f(c) h h
/ • h
Then lim [ f(c+h)- f(c)] h0
= f ’(c) • lim h h 0
= f ’(c) • 0 = 0
, and from here we get lim f(c+h) = f(c) h0
So lim [ f(c+h) - f(c)] = h0
So f is continuous at c for every c in (a,b).

24. The reverse of this theorem is not true.
Remark
The reverse of this theorem is not true.
Example: Since the derivative of f(x)= 5x2+x+1 is f ’(x) = 10x+1, which exists for every real number x. So f(x)= 5x2+x+1 is continuous everywhere.
Counter example: We know that f(x) = |x| is continuous on R , but at x=0 it’s not differentiable since:
lim l0+hl –l0l
h 0 h
= lim lhl h 0 h
+1 if h 0 –1 if h0
, which approaches to

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

26. Derivatives of some elementary functions
If the derivative of a function is its slope, then for a constant function, the derivative must be zero. There is no change...
example:
The derivative of a constant is zero.

27. Derivatives of some elementary functions
We saw that if ,
This is part of a pattern.
examples:
power rule

28. 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.)

29. 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. )

30. 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.

31. Derivatives of Trigonometric Functions
Proof

32. Derivative of the cosine Function
Find the derivative of cos x:

33. Derivative of the cosine function is sine (cont.)

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

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

36. The Derivatives of the Sum, Difference, Product and Quotient

37. Proof

41. HOMEWORK!! 

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

46. Remark
f(g(x))’= f ’(g(x)) g’(x) says that to get the derivative of the “nested functions” you multiply the derivative of each one starting from left to right and so on

47. Example for using Chain rule
Example : Find y’(1) for y = (3x2-2)3( 5x3-x-3)4
y ’= 3(3x2 -2)2 (3x2-2)’( 5x3-x-3)4 + (3x2-2)3 4( 5x3-x-3)3 ( 5x3-x-3)’
y ’= 3(3x2 -2)2 (6x) ( 5x3-x-3)4 + (3x2-2)3 4( 5x3-x-3)3 (15x2-1)
y ’(1) = 3(3-2)2 (6) (5-1-3) (3-2) (5-1-3)3 (15-1)
YOUR TURN!
= 74
dy dx
. 2 x √5x2+4
, find when x=
For y =

48. Example for using Chain rule

49. Example for using Chain rule
The chain rule can be used more than once.
(That’s what makes the “chain” in the “chain rule”!)

50. 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.

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

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

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

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

55. Implicit Differentiation
Find if
Substitute
back into the equation.

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

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

58. Derivatives of Inverse Trigonometric Functions

59. Derivatives of Inverse Trigonometric Functions
Find

60. 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

61. 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.

62. 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

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

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

65. Derivatives of Exponential and Logarithmic Functions

66. 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:

67. Derivatives of Exponential and Logarithmic Functions

68. 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