1314 Math Lesson 6 The Chain Rule In this lesson, you will learn the last of the basic rules for finding derivatives, the chain rule. Example 1: Decompose h( x) = 3 x 2 5 x + 6 into functions f (x) and g (x) such that h( x) = ( f g )( x).
(
)
4
Rule 10: The Chain Rule
d [ f (g (x ))] = f ' (g ( x) )g ' ( x) dx
Example 2: Find the derivative if f ( x) = 3 x 3 4 .
(
)
5
Example 3: Find the derivative if f ( x) = 4 x 4 10 .
Example 4: Find the derivative if f ( x) =
(2 x
5
2
8
)
4
.
We can also apply the chain rule in problems involving the exponential function and the logarithmic function.
Rule 11: The Chain Rule (Exponential Function)
d f ( x) e = e f ( x ) f ' ( x) dx (Note, we will not work chain rule problems where the exponential function has a base other than e.)
[
]
Example 5: Find the derivative: f ( x) = 2e 5 x .
Rule 12: Chain The Rule (Logarithmic Function)
d [ln | f ( x) |] = f ' ( x) , provided f ( x) > 0 dx f ( x)
Example 6: Find the derivative: f ( x) = ln(3 x 2 + 5) .
Sometimes it is helpful to use the properties of logarithms to simplify a problem before we find the derivative:
Example 7: Find the derivative: f ( x) = ln 5 x 4
( )
Example 8: Find the derivative: f ( x) = ln x 2 + 1 x 3 5
[(
)(
3
)]
4
We can also use the chain rules together with either the product rule or the quotient rule.
Example 9: Find the...