The function below is the composition of two other functions. Without rewriting the original function, determine the inside and outside functions.
f(x) = cos(sin x)
Answer
sin x provides the input to cos (□), therefore sin x is the inside function and cos(□) is the outside function.
Example 2
The function below is the composition of two other functions. Without rewriting the original function, determine the inside and outside functions.
f(x) = ln (x3)
Answer
x3 is the input to ln (□), therefore x3 is the inside function and ln(□) is the outside function.
Example 3
The function below is the composition of two other functions. Without rewriting the original function, what are the inside and outside functions?
f(x) = 47x
Answer
We would first figure out 7x and then find 4 raised to that power, so 7x is the inside function and 4{□} is the outside function.
Example 4
The function below is the composition of two other functions. Without rewriting the original function, determine the inside and outside functions.
f(x) = (3x2 + 1)4
Answer
(3x2 + 1) is inside the function (□)4, therefore (3x2 + 1) is the inside function and (□)4 is the outside function.
Example 5
The function below is the composition of two other functions. Without rewriting the original function, what are 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 the input to our outside function,
e{□}.
Example 6
Find the derivative of the function using the chain rule.
h(x) = (x5 + 4)99
Answer
The outside function is
f(□) = (□)99
and the inside function is
g(x) = x5 + 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(x5 + 4)98.
The derivative of the inside function is
g ' (x) = 5x4,
so the final answer is
h ' (x) = f ' (g(x)) · g'(x) = 99(x5 + 4)98 · 5x4.
Example 7
What is h ' (x) for the following function?
h(x) = sin(ln x)
Answer
For h(x) = sin(ln x),
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
Example 8
What is h ' (x)?
h(x) = ln(ln x)
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
.
Example 9
What is the derivative of the following function?
h(x) = esinx
Answer
For
h(x) = esin 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
esin x.
Then we multiply by the derivative of the inside function, which is cos x, to reach the final answer:
h ' (x) = esin x · cos x.
Example 10
What's the derivative of the following function?
h(x) = ln x3
Answer
The outside function is
f(□) = ln (□)
and the inside function is
g(x) = x3.
Then
and
g ' (x) = 3x2,
therefore
We can simplify this to just .
Example 11
Let h(x) = (lnx)2. The following equation for h ' (x) comes from applying the chain rule incorrectly. Identify the mistake(s) in the equation.
Answer
For both parts of the problem, we'll 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.
The equation
is the derivative of f evaluated at the derivative of g, or
f ' (g'(x)).
That's definitely not what the chain rule tells us to do.
Example 12
Let h(x) = (lnx)2. The following equation for h ' (x) comes from applying the chain rule incorrectly. Identify the mistake(s) in the equation.
h ' (x) = 2(lnx)
Answer
h'(x) = 2(lnx) is equal to
f ' (g(x)).
This is partway to the correct answer, but it needs to be multiplied by g ' (x).
Example 13
Let h(x) = (lnx)2. The following equation for h ' (x) comes from applying the chain rule incorrectly. Identify the mistake(s) in the equation.
Answer
This is equal to
f(g ' (x)),
or the outside function evaluated at the derivative of the inside function, which is strange.