Question
Step 1
Recall the Chain Rule in terms of the functions
and
, which states that if
is differentiable at
and
is differentiable at
, then the composite function
o
defined by
is differentiable at
and
is given by the following.
We are given the function
To write this is the form
we can let
and
SUBMI
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Recall the Chain Rule in terms of the functions
SUB (YOU CANNOT COME BACK)
Need Help? Read It
SUBMIT ANSWER
SKI
Ask by Burns Bright. in the United States
Mar 16,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
To express
as
, let:
Applying the Chain Rule:
So, the derivative is
.
Solution
To express the function
in the form
, we need to identify the inner function
and the outer function
.
From the problem, we have already defined:
Now, we need to determine
. Since
is the sine of
, we can define
as follows:
Thus, we can write:
In summary:
Now, we can apply the Chain Rule to find the derivative
.
-
First, we need to find
and : -
-
Now, we can apply the Chain Rule:
Substituting the values we found:
Thus, the derivative of
is:
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Beyond the Answer
To express the function
in the form
, we indeed set
. Now, we need to find the function
. Notice that
should be the function that takes the output of
and provides us with
. Therefore, if we let
, then we can express
as
.
Don’t forget to check how the chain rule applies in this context! First, find the derivatives of both
and
:
- The derivative
. - The derivative
.
Now you can easily computeusing the chain rule!
Doing derivatives in calculus might seem a bit mystifying at first, but it’s ultimately about understanding how functions interact with one another! Think of it like following a recipe: you have your main ingredients (the functions) and you have to mix them together (taking derivatives) to create something delicious (the derivative of the composite function).