Question
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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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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 .
  1. First, we need to find and :
  2. 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 compute using 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).

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