1. 4. THE DERIVATIVE
(NPD)

2. Concept of Differentiation
Differentiation at one point
Introduction (two problems with one theme)
a. Tangent Lines
The secant line connecting P and Q has slope Q
f(x)
If x  c , then the secant line through P and Q will approach the tangent line at P. Thus, the slope of the tangent line is
f(x)-f(c) P
f(c)
x-c c x

3. b. Instantaneous Velocity
Let a particle travel around an axis and positition of particle at time t is s = f(t). If the particle has a coordinate f(c) at time c and f(c+h) at time c + h ,
thus the average velocity during the time interval [c, c+h] is
Elapsed time
Distance traveled s
f(c)
f(c+h) c
c+h

4. If h 0, we get instantaneous velocity at x = c :
Let x = c + h, instantaneous velocity can be written as
The slope of the tangent line and instantaneous
velocity have the same formula
Definition : The (first) derivative of f at x = c, denoted by is defined by
if limit exist

5. Other notation :
Example 1: Let Evaluate

7. Derivatives from the right and the left
Derivatives from the left at c, denoted by , is defined by
Derivatives from the right at c, denoted by , is defined by
A function f is said differentiable at c ( exist )
If and

8. Determine whether f(x) is differentiable at x = 1
Example 2: Let
Determine whether f(x) is differentiable at x = 1
If f is differentiable, find
Solution a. b.
Thus, f is differentiable at x = 1

9. Theorem If f is differentiable at point c  f is continuous at c.
Proof :
We will proof
Thus
The converse, however, is false, a function may be continuous at a point but not differentiable

10. We will show f(x)=|x| is continuous at x = 0
Example 3
Show that, f ( x ) = |x| is continuous at x = 0 but f(x) is not differentiable at x = 0
Solution
We will show f(x)=|x| is continuous at x = 0 
f(0) = 0  
Thus, f is continuous at x = 0

11. Because then f is not differentiable at x = 0
Determine whether f is differentiable at x = 0
Because then f is not differentiable at x = 0

12. When does a Function Not Have a Derivative at a Point?

14. Example 4: Find the values of a and b so that f(x) will be differentiable at x = 1.
Solution :
Function f(x) is differentiable at x = 1 if
a. f is continuous at x = 1 (necessary condition)
f is continuous at x = 1 if f is continuous from the left at x = 1 and f is continuous from the right, or

15. b. Derivatives from the left = derivatives from the right at c (sufficient condition)
So that : a = 2 and b = 1.

16. EXERCISES Determine whether is differentiable at x = 1
Determine whether is differentiable at real
Determine whether is differentiable at x = 2
Determine whether is differentiable at real
Find the values of a and b so that will be differentiable at x = 3

17. Rules for Finding Derivatives
Definition The derivative with respect to x of the function f (x), denoted by , defined by the formula
Or, if h = t-x
for which the limit exist.
Other Notation
Notation is called as Leibniz Notation.

18. By using the definition above, we have formula
1. If f (x) = k, then

19. Proof formula number 4
Let h(x) = f(x)g(x)

20. Example 5
1. Find of
solution :
2. Find of
Solution:
3. Find of
Solution :

21. Problems
Find of 1. 2. 3. 4. 5.

22. Derivative of Trigonometric Function
Proof:
a. Let f(x) = sin x, then

23. b. Let f(x) = cos x, then

24. The derivatives of remaining trigonometric functions can be obtained using formula derivative of u/v

25. The Chain Rule Let y = f(u) and u = g(x). If and exists, then
Example 6: Find if
Solution :
Let so that
because of
and
then

26. Let y = f(u), u = g(v), v = h(x), if exists, then
Example 7: Find if
Solution
Let
u = Sin v
So that

27. Or, in easy way

28. Problems
Find dy/dx 1. 2. 3. 4. 5. 6.
y = sin x tan [ x2 + 1 ]

29. The Higher Derivative
The nth derivative of f(x) is the derivative of (n-1)th of f(x).
The first derivative
The second derivative
The third derivative
The nth derivative
Example 8: Find if
Solution :

30. A. Find the second derivative of
Problems
A. Find the second derivative of 1. 2. 3. 4.
B. Find the value of c so that if
C. Find the values of a, b and c if
, g (1) = 5,
and

31. Implicit Differentiation
Function defined explicitly
Function defined implicitly

32. Example 9: By implicit differentiation, find dy/dx if
Answer:

34. Problems
Find by implicit differentiation 1. 2. 3. 4.

35. Tangent Line and Normal Line
Tangent line at point (x0,y0)
with at point (x0,y0)
Normal Line and tangent line perpendicular each other.
Normal line at point (x0,y0)
with where at point (x0,y0)

36. Tangent Line and Normal Line

37. Example 10:
Find the tangent line and normal line at point (2,6) of
Answer :
The tangent line at point (2,6) :
The normal line at point (2,6) :

38. Example 11: Find the tangent line and normal line at point x = 1 of
Answer :
If x = 1 is substituted in equation we found
Thus, we get points (1,3) and (1,-2)
By implicit differentiation

39. At point (1,3)
At point (1,-2)
The tangent line
The tangent line
The normal line
The normal line

40. Example 12

41. Exercises
1. Let
Find tangent line and normal line at point (2,1)
2. Let
Find tangent line and normal line at point