1. Derivatives: definition and derivatives of various functions

2. What is a derivative?
A function
the rate of change of a function
the slope of the line tangent to the curve

3. The tangent line
single point
of intersection

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

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

6. closer and closer… a

7. watch the slope...

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

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

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

11. 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!

12. The tangent line problem
secant line
(x, f(x))
(x, f(x)) is the point of tangency and
f(x ) – f(x) x
is a second point on the graph of f.

13. The slope between these two points is
Definition of Tangent Line with Slope m

14. Find the slope of the graph of f(x) = x2 +1 at
the point (-1,2). Then, find the slope of the
tangent line.
(-1,2)

15. f(x) = x2 + 1
Therefore, the slope
at any point (x, f(x))
is given by m = 2x
What is the slope
at the point (-1,2)?
m = -2

16. The limit used to define the slope of a tangent
line is also used to define one of the two funda-
mental operations of calculus --- differentiation
Definition of the Derivative of a Function
f’(x) is read “f prime of x”

17. Find f’(x) for f(x) = and use the result to find
the slope of the graph of f at the points (1,1) & (4,2). What happens at the point (0,0)? 1

18. Derivative of a Function
“the derivative of f with respect to x”
“y prime”
“the derivative of y with respect to x”
“the derivative of f of x”

19. Differentiability.
To be differentiable, a function must be continuous and smooth.
Derivatives will not exist at the following:
cusp
corner
vertical tangent
discontinuity

20. Differentiability: Intermediate value theorem
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 .

21. Rules of differentiation

22. Differentiation: Function Types

23. Differentiation: Function Types

24. Differentiation: Function Types

25. Rules of Differentiation

26. Rules of Differentiation

27. Rules of Differentiation

28. Rules of Differentiation

29. Examples of Composite Functions

30. The Chain Rule of Composite Functions

31. Examples

32. Session Problem Set