1. Lesson: Derivative Basics - 2
Obj – The Derivative Function

3. Differentiation – To find the derivative
of a function.
Derivative – The slope of the tangent
line of a curve.
- Instantaneous rate of change
- y’, f’(x), dy/dx, d/dx

4. Ex. 1: Find the derivative, with respect to x,
of f(x) = x2 + 1and use it to find the
equation of the tangent line at x = 2.

5. Differentiability:
It is possible that the limit that defines a
derivative of a function, f, may not exist
at certain points in the domain of f.
When this occurs, the derivative at that
point is undefined.
If the derivative does exist at a point, x0,
then the function is differentiable at x0.

6. In general, a function is differentiable at a point
if the graph has a unique tangent line at the point.
In other words, to be differentiable means that it
has a derivative.

7. Conditions when a function is not differentiable.
Corner Points
Points of Vertical Tangency
Cusps
Why?
At corner points, the slopes of the secant lines
have different values from the right & left sides.
Points of vertical tangency have slopes of secant
lines with undefined slopes.
Cusps have non equal slopes of on opposing
sides.

9. Ex. 2: Given the graph of f(x) below, find
the value of:

10. Recall that when finding , that you are
Finding the derivative of the function, which is
actually just the slope of the tangent line.
So when you are asked to find , all
you need to do is find the slope of the line
tangent to the curve at x = -2.
Draw a tangent line on your graph
Find the slope: rise/run

11. Solution:
Tangent line at x = -2 has a slope of 0.

12. Solution: Tangent line at x = -1 has a rise of about 2
And a run of about 1, so f ’(-1) = 2/1 = 2

13. Solution: Tangent line at x = -1 has a rise of about 1
And a run of about -2, so f ’(2) = 1/-2

14. Ex. 3: Match each graph in # a - f with the graph of
its derivative, from A - F.

16. Example 3 solutions
a - D
b - F
c - B
d - C
e - A
f - E

17. Theorem 3.3.1
The derivative of a constant is zero
d/dx [c] = 0
Ex. If f(x) = 7, then
If g(x) = -8, then

18. The basis for the Power Rule
Or my personal favorite version:
The basis for the Power Rule
In words: The derivative of a term of a function
is the product of its exponent and
coefficient, along with the variable
raised to one less power.

19. Examples:

20. EX. 4: Find each derivativeb)
c) d)
e) f)

21. Solutions to EX. 4: b) c)

22. EX. 4: Find each derivative

23. Examples:

24. EX. 6: An object is dropped from
the Empire State Building that is
modeled by the equation
h(t) = t2, where h(t) is
measured in feet and t is in sec.
Find the velocity function of the object.
Find the instantaneous velocity at t = 5.
Find the time interval over which the
function is valid.
D. What is the velocity of the object
as it hits the ground?

25. A. B.

26. C. To determine the time interval for
which the function is valid, we must
examine possible values of the domain
that makes sense. (The time the
object is dropped and the time it hits
the ground).
The initial time is easy (t = 0).
To find the final time, we need to locate
when the object hits the ground. (when
h(t) = 0)

28. So the function h(t)= 1250 – 16t2 is
valid for the time interval [0,8.84]
D. One could assume that the instantaneous
velocity of the object when it hits the
ground is zero, but we know the dangers
of assuming, don’t we? The velocity is
not zero until after it hits the ground.
We need to examine the value from our
velocity function when the object hits
the ground.

30. :EX. 7: Let f(x) = 2x6 + x-9
Find
EX. 8: Find dy/dx if
EX. 9: If
Find

31. EX. 10: At what points does the graph of have a horizontal tangent
line?
What do we know about the slope of a
horizontal line?
How can we use that information
to find what we are looking for?

34. After substituting those points into
the original equation, we get:
and
Therefore horizontal tangents exist
at the points (1, 2) and (-1, 6).

35. EX. 11: If
Find:

36. Normal line – The line perpendicular to the
tangent line, at the point of
tangency.
Normal Line
Tangent Line

37. Normal Line
Tangent Line
Secant Line

38. Ex. 12: Write the equation of the tangent
line and the normal line to the
function f(x) = 2x2 + 3x +5, at the
point (-1, 4)