When dealing with a composition
h(x) = f(g(x)),
the function f is called the outside function and the function g is called the inside function. f is the outside function because it's written on the outside of the parentheses:
f(g(x))
g is the inside function because it's written on the inside of the parentheses:
f(g(x)).
Written like this, the idea of outside and inside functions makes a lot of sense, but somehow it always gets more complicated when we start looking at actual functions.
"Outside" and "Inside" Functions
Another way to think of outside and inside functions in the case of the composition
h(x) = f(g(x))
is to think about what we need to do in order to evaluate h at a particular value of x. First we would need to find the output of g at that value of x. Then we would put that output into f. The inside function creates the input that we need for the outside function. Like the product and quotient rules, the outside function is the one we evaluate last.
Sample Problem
Look at the function
h(x) = e^{{sin x}}.
This is a composition of the function e^{{(□)}} and sin x. In order to evaluate this function at , the first thing we do
is find the output of sin at this value of x:
The next thing we do is find e raised to that power:
For this composition, sin x is the inside function because sin x is the input to e^{{(□)}}. Therefore e^{{(□)}} must be the outside function. We're writing
e^{{(□)}}
instead of
e^{x}
to emphasize that we'll be raising e to some other power besides x.
The chain rule states that
(f(g(x)))' = f '(g(x)) × g'(x).
If we state the chain rule with words instead of symbols, it says this: to find the derivative of the composition f(g(x)),
- identify the outside and inside functions
- find the derivative of the outside function and then use the original inside function as the input
- multiply by the derivative of the inside function.
Of course, we also want to simplify the answer into something reasonable.
Some people like thinking about the chain rule as
f '(g(x)) × g'(x),
while some prefer
(derivative of outside function evaluated at inside function)(derivative of inside function).
Some prefer Leibniz notation. Use the way that works best.
Be Careful: When using the chain rule, use the original inside function as input to the derivative of the outside function. Then remember to multiply by the derivative of the inside function!
We can use the chain rule on functions that are nested 3 or more deep; we need to write things down carefully.
Practice:
Let Without rewriting f, identify the inside and outside functions. | |
If we were to evaluate this function on a calculator, first we would need to find x^{2}, therefore x^{2} is the inside function. x^{2} provides the input to the function therefore is the outside function. | |
Find the derivative of h(x) = cos{sin x}.
| |
We can think of this function as f(g(x)), where f(□) = cos(□) and { g(x)} = { sin x}. Then f'(□) = -sin (□), therefore f'({ g(x)}) = -sin({ sin x}). Multiplying by g'(x) = cos x, we find h'(x) = f'(g(x)) × g'(x) = -sin(sin x) × (cos x). | |
Find the derivative of h(x) = sin (x^{2}).
| |
- First we identify the outside and inside functions. The outside function is the one on the outside:
sin(□) and the inside function is the one on the inside: { x^{2}}. - The derivative of the outside function is
cos (□). Using the inside function { x^{2}} as input, we find cos({ x^{2}}). - Finally, we multiply by the derivative of the inside function. The derivative of the inside function is
(x^{2})' = 2x, the final answer to the problem is h'(x) = cos({ x^{2}}) × 2x. | |
Find the derivative of the function h(x) = (ln x)^{2}.
| |
The outside function is f(□) = (□)^{2} and the inside function is { g(x)} = { ln x}. The derivative of the outside function is f'(□) = 2(□), and if we use the inside function as input we find f'({ g(x)}) = 2({ ln x}). The derivative of the inside function is multiplying the appropriate things together we find our final answer:
| |
The function below is the composition of two other functions. Without rewriting the original function, determine the inside and outside functions.
Answer
- sin x provides the input to cos (□), therefore sin x is the inside function and cos(□) is the outside function.
The function below is the composition of two other functions. Without rewriting the original function, determine the inside and outside functions.
Answer
- x^{3} is the input to ln (□), therefore x^{3} is the inside function and ln(□) is the outside function.
The function below is the composition of two other functions. Without rewriting the original function, determine the inside and outside functions.
Answer
- We would first figure out 7x and then find 4 raised to that power. 7x is the inside function and 4^{{□}} is the outside function.
The function below is the composition of two other functions. Without rewriting the original function, determine the inside and outside functions.
Answer
- (3x^{2} + 1) is inside the function (□)^{4}, therefore (3x^{2} + 1) is the inside function and (□)^{4} is the outside function.
The function below is the composition of two other functions. Without rewriting the original function, determine the inside and outside functions.
- f(x) =
Answer
- This is like our first example. is the inside function because we need to compute this first. Then we use this as input to our outside function,
e^{{□}}.
Find the derivative of the function. These do all require the chain rule.
- h(x) = (x^{5} + 4)^{{99}}
Answer
f(□) = (□)^{{99}}
and the inside function is
{ g(x)} = { x^{5} + 4}.
The derivative of the outside function is
f'(□) = 99(□)^{{98}},
and the derivative of the outside function with the inside function as input is
f'({ g(x)}) = 99({ x^{5} + 4})^{{98}}.
The derivative of the inside function is
g'(x) = 5x^{4},
the final answer is
h'(x) = f'(g(x)) × g'(x) = 99(x^{5} + 4)^{{98}} × 5x^{4}.
Find the derivative of the function. These do all require the chain rule.
Answer
the outside function is sin (□) and the inside function is { ln x}.The derivative of the outside function is
cos(□)
and the derivative of the inside function is
Taking the derivative of the outside function evaluated at the inside function, and multiplying by the derivative of the inside function, we find
Find the derivative of the function. These do all require the chain rule.
Answer
- We can think of the function
h(x) = ln(ln x)
as
h(x) = f(g(x))
where both
f(□) = ln(□)
and
{ g(x)} = { ln x}.
Then
and
therefore
Find the derivative of the function. These do all require the chain rule.
Answer
h(x) = e^{{ sin x}}
the outside function is
e^{{□}}
and the inside function is
sin x.
The derivative of the outside function is also
e^{{□}},
and using the inside function for input gets us
e^{sin x}.
Then we multiply by the derivative of the inside function, which is cos x, to reach the final answer:
h'(x) = e^{{sin x}} × cos x.
Find the derivative of the function. These do all require the chain rule.
Answer
f(□) = ln (□)
and the inside function is
{ g(x)} = { x^{3}}.
Then
and
g'(x) = 3x^{2},
therefore
Let h = (ln x)^{2}. The following formula for h'(x) comes from applying the chain rule incorrectly. Identify the mistake(s) in the formula.
Answer
For both parts of the problem, we will need to know the outside and inside functions and their derivatives.The outside function is
f(□) = (□)^{2}
and its derivative is
f'(□) = 2(□).
The inside function is
{ g(x)} = { ln x}
and its derivative is
- Since we know f, f', g, and g', we can see how the incorrect formulas were constructed.
is the derivative of f evaluated at the derivative of g, or
f'(g'(x)).
Let h = (ln x)^{2}. The of the following formulas for h'(x) comes from applying the chain rule incorrectly. Identify the mistake(s) in the formula.
Answer
- h'(x) = 2(ln x) is equal to
f'({ g(x)}).
This is partway to the correct answer, but it needs to be multiplied by g'(x).
Let h = (ln x)^{2}. The following formula for h'(x) comes from applying the chain rule incorrectly. Identify the mistake(s) in the formula.
Answer
This is equal to
f(g'(x)),
or the outside function evaluated at the derivative of the inside function, which is strange!
Let h = (ln x)^{2}.
- Find a correct formula for h'(x).
Answer
- Using the chain rule correctly,