The document discusses finding derivatives of trigonometric functions. It shows that the derivative of sin(x) is cos(x) and the derivative of cos(x) is -sin(x). It then uses these results and the chain rule to derive the derivatives of other trig functions like sin^2(x) and tan(x).

3. Derivatives of Trigonometric Functions

4. Derivatives of Trigonometric Functions
In part 1, we showed that:

5. Derivatives of Trigonometric Functions
In part 1, we showed that:
d
dx
(sin x) = cos x

6. Derivatives of Trigonometric Functions
In part 1, we showed that:
d
dx
(sin x) = cos x
Let’s now find the derivative of f (x) = cos x.

7. Derivatives of Trigonometric Functions
In part 1, we showed that:
d
dx
(sin x) = cos x
Let’s now find the derivative of f (x) = cos x.
There are two ways to proceed:

8. Derivatives of Trigonometric Functions
In part 1, we showed that:
d
dx
(sin x) = cos x
Let’s now find the derivative of f (x) = cos x.
There are two ways to proceed:
1. Apply the definition of the derivative.

9. Derivatives of Trigonometric Functions
In part 1, we showed that:
d
dx
(sin x) = cos x
Let’s now find the derivative of f (x) = cos x.
There are two ways to proceed:
1. Apply the definition of the derivative.
2. Apply the chain rule.

10. Derivatives of Trigonometric Functions
In part 1, we showed that:
d
dx
(sin x) = cos x
Let’s now find the derivative of f (x) = cos x.
There are two ways to proceed:
1. Apply the definition of the derivative.
2. Apply the chain rule.

11. Derivatives of Trigonometric Functions
So, let’s consider the identity:

12. Derivatives of Trigonometric Functions
So, let’s consider the identity:
f (x) = cos x = 1 − sin2
x

13. Derivatives of Trigonometric Functions
So, let’s consider the identity:
f (x) = cos x = 1 − sin2
x
We can apply the chain rule directly:

14. Derivatives of Trigonometric Functions
So, let’s consider the identity:
f (x) = cos x = 1 − sin2
x
We can apply the chain rule directly:
f (x) =
1
2
.

15. Derivatives of Trigonometric Functions
So, let’s consider the identity:
f (x) = cos x = 1 − sin2
x
We can apply the chain rule directly:
f (x) =
1
2
.
1
1 − sin2
x
.

16. Derivatives of Trigonometric Functions
So, let’s consider the identity:
f (x) = cos x = 1 − sin2
x
We can apply the chain rule directly:
f (x) =
1
2
.
1
1 − sin2
x
. (−2 sin x) .

17. Derivatives of Trigonometric Functions
So, let’s consider the identity:
f (x) = cos x = 1 − sin2
x
We can apply the chain rule directly:
f (x) =
1
2
.
1
1 − sin2
x
. (−2 sin x) . cos x

18. Derivatives of Trigonometric Functions
So, let’s consider the identity:
f (x) = cos x = 1 − sin2
x
We can apply the chain rule directly:
f (x) =
1
2
.
1
1 − sin2
x
. (−2 sin x) . cos x
= −
2 sin x cos x
2 1 − sin2
x

19. Derivatives of Trigonometric Functions
So, let’s consider the identity:
f (x) = cos x = 1 − sin2
x
We can apply the chain rule directly:
f (x) =
1
2
.
1
1 − sin2
x
. (−2 sin x) . cos x
= −
¡2 sin x cos x
¡2 1 − sin2
x

20. Derivatives of Trigonometric Functions
So, let’s consider the identity:
f (x) = cos x = 1 − sin2
x
We can apply the chain rule directly:
f (x) =
1
2
.
1
1 − sin2
x
. (−2 sin x) . cos x
= −
¡2 sin x cos x
¡2$$$$$$$Xcos x
1 − sin2
x

21. Derivatives of Trigonometric Functions
So, let’s consider the identity:
f (x) = cos x = 1 − sin2
x
We can apply the chain rule directly:
f (x) =
1
2
.
1
1 − sin2
x
. (−2 sin x) . cos x
= −
¡2 sin x cos x
¡2$$$$$$$Xcos x
1 − sin2
x
= −
sin x cos x
cos x
=

22. Derivatives of Trigonometric Functions
So, let’s consider the identity:
f (x) = cos x = 1 − sin2
x
We can apply the chain rule directly:
f (x) =
1
2
.
1
1 − sin2
x
. (−2 sin x) . cos x
= −
¡2 sin x cos x
¡2$$$$$$$Xcos x
1 − sin2
x
= −
sin x$$$cos x
$$$cos x
=

23. Derivatives of Trigonometric Functions
So, let’s consider the identity:
f (x) = cos x = 1 − sin2
x
We can apply the chain rule directly:
f (x) =
1
2
.
1
1 − sin2
x
. (−2 sin x) . cos x
= −
¡2 sin x cos x
¡2$$$$$$$Xcos x
1 − sin2
x
= −
sin x$$$cos x
$$$cos x
= − sin x
= −
sin x$$$cos x
$$$cos x
= − sin x

24. Derivatives of Trigonometric Functions
So, we have that:

25. Derivatives of Trigonometric Functions
So, we have that:
d
dx
(cos x) = − sin x

26. Derivatives of Trigonometric Functions
So, we have that:
d
dx
(cos x) = − sin x

27. Derivatives of Trigonometric Functions
So, we have that:
d
dx
(cos x) = − sin x
We can now solve some problems.

28. Example 1
Let’s find the derivative of the function:

29. Example 1
Let’s find the derivative of the function:
f (x) = sin2
x

30. Example 1
Let’s find the derivative of the function:
f (x) = sin2
x
We’ve already found this derivative when we calculated the
derivative of cos x.

31. Example 1
Let’s find the derivative of the function:
f (x) = sin2
x
We’ve already found this derivative when we calculated the
derivative of cos x.
We apply the chain rule:

32. Example 1
Let’s find the derivative of the function:
f (x) = sin2
x
We’ve already found this derivative when we calculated the
derivative of cos x.
We apply the chain rule:
f (x) = 2 sin x
d
dx
(sin x)

33. Example 1
Let’s find the derivative of the function:
f (x) = sin2
x
We’ve already found this derivative when we calculated the
derivative of cos x.
We apply the chain rule:
f (x) = 2 sin x
d
dx
(sin x) = 2 sin x cos x

34. Example 1
Let’s find the derivative of the function:
f (x) = sin2
x
We’ve already found this derivative when we calculated the
derivative of cos x.
We apply the chain rule:
f (x) = 2 sin x
d
dx
(sin x) = 2 sin x cos x

35. Example 2

36. Example 2
f (x) = tan x

37. Example 2
f (x) = tan x
Here we can use the product rule:

38. Example 2
f (x) = tan x
Here we can use the product rule:
f (x) =
sin x
cos x
=

39. Example 2
f (x) = tan x
Here we can use the product rule:
f (x) =
sin x
cos x
=
f (x) =
sin x
cos x
= sin x. (cos x)−1

40. Example 2
f (x) = tan x
Here we can use the product rule:
f (x) =
sin x
cos x
=
f (x) =
sin x
cos x
= sin x. (cos x)−1
f (x) =

41. Example 2
f (x) = tan x
Here we can use the product rule:
f (x) =
sin x
cos x
=
f (x) =
sin x
cos x
= sin x. (cos x)−1
f (x) = sin x.

42. Example 2
f (x) = tan x
Here we can use the product rule:
f (x) =
sin x
cos x
=
f (x) =
sin x
cos x
= sin x. (cos x)−1
f (x) = sin x. (cos x)−1
+

43. Example 2
f (x) = tan x
Here we can use the product rule:
f (x) =
sin x
cos x
=
f (x) =
sin x
cos x
= sin x. (cos x)−1
f (x) = sin x. (cos x)−1
+
d
dx
(sin x) . (cos x)−1

44. Example 2
f (x) = tan x
Here we can use the product rule:
f (x) =
sin x
cos x
=
f (x) =
sin x
cos x
= sin x. (cos x)−1
f (x) = sin x. (cos x)−1
+
d
dx
(sin x) . (cos x)−1
= sin x.

45. Example 2
f (x) = tan x
Here we can use the product rule:
f (x) =
sin x
cos x
=
f (x) =
sin x
cos x
= sin x. (cos x)−1
f (x) = sin x. (cos x)−1
+
d
dx
(sin x) . (cos x)−1
= sin x.(−1). (cos x)−2
.
d
dx
(cos x) +

46. Example 2
f (x) = tan x
Here we can use the product rule:
f (x) =
sin x
cos x
=
f (x) =
sin x
cos x
= sin x. (cos x)−1
f (x) = sin x. (cos x)−1
+
d
dx
(sin x) . (cos x)−1
= sin x.(−1). (cos x)−2
.
d
dx
(cos x) + cos x. (cos x)−1

47. Example 2
f (x) = tan x
Here we can use the product rule:
f (x) =
sin x
cos x
=
f (x) =
sin x
cos x
= sin x. (cos x)−1
f (x) = sin x. (cos x)−1
+
d
dx
(sin x) . (cos x)−1
= sin x.(−1). (cos x)−2
.
¨¨
¨¨¨¨B− sin x
d
dx
(cos x) + cos x. (cos x)−1

48. Example 2
f (x) = tan x
Here we can use the product rule:
f (x) =
sin x
cos x
=
f (x) =
sin x
cos x
= sin x. (cos x)−1
f (x) = sin x. (cos x)−1
+
d
dx
(sin x) . (cos x)−1
= sin x.(−1). (cos x)−2
.
¨¨¨
¨¨¨B− sin x
d
dx
(cos x) + cos x. (cos x)−1
=
sin2
x.
cos2 x
+

49. Example 2
f (x) = tan x
Here we can use the product rule:
f (x) =
sin x
cos x
=
f (x) =
sin x
cos x
= sin x. (cos x)−1
f (x) = sin x. (cos x)−1
+
d
dx
(sin x) . (cos x)−1
= sin x.(−1). (cos x)−2
.
¨¨¨
¨¨¨B− sin x
d
dx
(cos x) + cos x. (cos x)−1
=
sin2
x.
cos2 x
+
cos x
cos x

50. Example 2
f (x) = tan x
Here we can use the product rule:
f (x) =
sin x
cos x
=
f (x) =
sin x
cos x
= sin x. (cos x)−1
f (x) = sin x. (cos x)−1
+
d
dx
(sin x) . (cos x)−1
= sin x.(−1). (cos x)−2
.
¨¨¨
¨¨¨B− sin x
d
dx
(cos x) + cos x. (cos x)−1
=
sin2
x.
cos2 x
+
&
&
&b
1
cos x
cos x

51. Example 2
f (x) = tan x
Here we can use the product rule:
f (x) =
sin x
cos x
=
f (x) =
sin x
cos x
= sin x. (cos x)−1
f (x) = sin x. (cos x)−1
+
d
dx
(sin x) . (cos x)−1
= sin x.(−1). (cos x)−2
.
¨¨¨
¨¨¨B− sin x
d
dx
(cos x) + cos x. (cos x)−1
=
sin2
x.
cos2 x
+
&
&
&b
1
cos x
cos x
= tan2
x + 1

52. Example 2
f (x) = tan x
Here we can use the product rule:
f (x) =
sin x
cos x
=
f (x) =
sin x
cos x
= sin x. (cos x)−1
f (x) = sin x. (cos x)−1
+
d
dx
(sin x) . (cos x)−1
= sin x.(−1). (cos x)−2
.
¨¨¨
¨¨¨B− sin x
d
dx
(cos x) + cos x. (cos x)−1
=
sin2
x.
cos2 x
+
&
&
&b
1
cos x
cos x
= tan2
x + 1