In this presentation we learn to apply the chain rule to solve problems. For more lessons: http://www.intuitive-calculus.com/chain-rule.html

3. The Chain Rule
Given a composite function:

4. The Chain Rule
Given a composite function:
h(x) = f [g(x)]

5. The Chain Rule
Given a composite function:
h(x) = f [g(x)]
The chain rule says that:

6. The Chain Rule
Given a composite function:
h(x) = f [g(x)]
The chain rule says that:
h (x) = f [g(x)] g (x)

7. The Chain Rule
Given a composite function:
h(x) = f [g(x)]
The chain rule says that:
h (x) = f [g(x)] g (x)
Let’s consider again:

8. The Chain Rule
Given a composite function:
h(x) = f [g(x)]
The chain rule says that:
h (x) = f [g(x)] g (x)
Let’s consider again:
h(x) = sin x2

9. The Chain Rule
Given a composite function:
h(x) = f [g(x)]
The chain rule says that:
h (x) = f [g(x)] g (x)
Let’s consider again:
h(x) = sin
outer function
x2

10. The Chain Rule
Given a composite function:
h(x) = f [g(x)]
The chain rule says that:
h (x) = f [g(x)] g (x)
Let’s consider again:
h(x) = sin x2

11. The Chain Rule
Given a composite function:
h(x) = f [g(x)]
The chain rule says that:
h (x) = f [g(x)] g (x)
Let’s consider again:
h(x) = sin x2
inner function

12. The Chain Rule
Given a composite function:
h(x) = f [g(x)]
The chain rule says that:
h (x) = f [g(x)] g (x)
Let’s consider again:
h(x) = sin x2

13. The Chain Rule
Given a composite function:
h(x) = f [g(x)]
The chain rule says that:
h (x) = f [g(x)] g (x)
Let’s consider again:
h(x) = sin x2
h (x) = cos x2
.

14. The Chain Rule
Given a composite function:
h(x) = f [g(x)]
The chain rule says that:
h (x) = f [g(x)] g (x)
Let’s consider again:
h(x) = sin x2
h (x) = cos x2
derivative of outer function
.

15. The Chain Rule
Given a composite function:
h(x) = f [g(x)]
The chain rule says that:
h (x) = f [g(x)] g (x)
Let’s consider again:
h(x) = sin x2
h (x) = cos x2
.

16. The Chain Rule
Given a composite function:
h(x) = f [g(x)]
The chain rule says that:
h (x) = f [g(x)] g (x)
Let’s consider again:
h(x) = sin x2
h (x) = cos x2
.2x

17. The Chain Rule
Given a composite function:
h(x) = f [g(x)]
The chain rule says that:
h (x) = f [g(x)] g (x)
Let’s consider again:
h(x) = sin x2
h (x) = cos x2
. 2x
derivative of inner function

18. The Chain Rule
Given a composite function:
h(x) = f [g(x)]
The chain rule says that:
h (x) = f [g(x)] g (x)
Let’s consider again:
h(x) = sin x2
h (x) = cos x2
.2x

19. Example 1

20. Example 1
f (x) = 3
√
sin x

21. Example 1
f (x) = 3
√
sin x
We can write it as:

22. Example 1
f (x) = 3
√
sin x
We can write it as:
f (x) = 3 (sin x)
1
2

23. Example 1
f (x) = 3
√
sin x
We can write it as:
f (x) = 3

 sin x
inner function


1
2

24. Example 1
f (x) = 3
√
sin x
We can write it as:
f (x) = 3 $$$Xy
sin x
1
2

25. Example 1
f (x) = 3
√
sin x
We can write it as:
f (x) = 3 $$$Xy
sin x
1
2
= 3y
1
2

26. Example 1
f (x) = 3
√
sin x
We can write it as:
f (x) = 3 $$$Xy
sin x
1
2
= 3y
1
2
We take the derivative of the outer function:

27. Example 1
f (x) = 3
√
sin x
We can write it as:
f (x) = 3 $$$Xy
sin x
1
2
= 3y
1
2
We take the derivative of the outer function:
f (x) = 3.
1
2
y− 1
2 .

28. Example 1
f (x) = 3
√
sin x
We can write it as:
f (x) = 3 $$$Xy
sin x
1
2
= 3y
1
2
We take the derivative of the outer function:
f (x) = 3.
1
2
y−1
2 .y

29. Example 1
f (x) = 3
√
sin x
We can write it as:
f (x) = 3 $$$Xy
sin x
1
2
= 3y
1
2
We take the derivative of the outer function:
f (x) = 3.
1
2
y−1
2 .y =
3
2
(sin x)−1
2 . (sin x)

30. Example 1
f (x) = 3
√
sin x
We can write it as:
f (x) = 3 $$$Xy
sin x
1
2
= 3y
1
2
We take the derivative of the outer function:
f (x) = 3.
1
2
y−1
2 .y =
3
2
(sin x)−1
2 . (sin x) =
3
2
(sin x)−1
2 . cos x

31. Example 1
f (x) = 3
√
sin x
We can write it as:
f (x) = 3 $$$Xy
sin x
1
2
= 3y
1
2
We take the derivative of the outer function:
f (x) = 3.
1
2
y−1
2 .y =
3
2
(sin x)−1
2 . (sin x) =
3
2
(sin x)−1
2 . cos x
f (x) =
3 cos x
√
sin x

32. Example 1
f (x) = 3
√
sin x
We can write it as:
f (x) = 3 $$$Xy
sin x
1
2
= 3y
1
2
We take the derivative of the outer function:
f (x) = 3.
1
2
y−1
2 .y =
3
2
(sin x)−1
2 . (sin x) =
3
2
(sin x)−1
2 . cos x
f (x) =
3 cos x
√
sin x

33. Example 2

34. Example 2
f (x) = a
√
sin 2x

35. Example 2
f (x) = a
√
sin 2x = a (sin 2x)
1
2

36. Example 2
f (x) = a
√
sin 2x = a (sin 2x)
1
2
f (x) = a.
1
2
. (sin 2x)−1
2 .

37. Example 2
f (x) = a
√
sin 2x = a (sin 2x)
1
2
f (x) = a.
1
2
. (sin 2x)−1
2 . (sin 2x)

38. Example 2
f (x) = a
√
sin 2x = a (sin 2x)
1
2
f (x) = a.
1
2
. (sin 2x)−1
2 . (sin 2x)
f (x) = a.
1
2
. (sin 2x)−1
2 . cos 2x.

39. Example 2
f (x) = a
√
sin 2x = a (sin 2x)
1
2
f (x) = a.
1
2
. (sin 2x)−1
2 . (sin 2x)
f (x) = a.
1
2
. (sin 2x)−1
2 . cos 2x.2

40. Example 2
f (x) = a
√
sin 2x = a (sin 2x)
1
2
f (x) = a.
1
2
. (sin 2x)−1
2 . (sin 2x)
f (x) = a.
1
¡2
. (sin 2x)−1
2 . cos 2x.¡2

41. Example 2
f (x) = a
√
sin 2x = a (sin 2x)
1
2
f (x) = a.
1
2
. (sin 2x)−1
2 . (sin 2x)
f (x) = a.
1
¡2
. (sin 2x)−1
2 . cos 2x.¡2
f (x) =
a cos 2x
√
sin 2x

42. Example 2
f (x) = a
√
sin 2x = a (sin 2x)
1
2
f (x) = a.
1
2
. (sin 2x)−1
2 . (sin 2x)
f (x) = a.
1
¡2
. (sin 2x)−1
2 . cos 2x.¡2
f (x) =
a cos 2x
√
sin 2x