Math Problem Statement
For the following composite function, find an inner function uequalsg(x) and an outer function yequalsf (u) such that yequalsf(g(x)). Then calculate StartFraction dy Over dx EndFraction . yequalscosine StartFraction 8 x Over 5 EndFraction Question content area bottom Part 1 Select the correct choice below and fill in the answer box to complete your choice. A. StartFraction dy Over dx EndFraction equalsStartFraction d Over du EndFraction (u)timesStartFraction d Over dx EndFraction left parenthesis StartFraction 8 x Over 5 EndFraction right parenthesis equals enter your response here B. StartFraction dy Over dx EndFraction equalsStartFraction d Over du EndFraction left parenthesis StartFraction 8 u Over 5 EndFraction right parenthesis timesStartFraction d Over dx EndFraction (cosine x)equals enter your response here C. StartFraction dy Over dx EndFraction equalsStartFraction d Over du EndFraction (cosine u)timesStartFraction d Over dx EndFraction left parenthesis StartFraction x Over 5 EndFraction right parenthesis equals enter your response here D. StartFraction dy Over dx EndFraction equalsStartFraction d Over du EndFraction (cosine u)timesStartFraction d Over dx EndFraction left parenthesis StartFraction 8 x Over 5 EndFraction right parenthesis equals
Solution
To solve the problem, we need to express the given function as a composite function of two functions, find the derivative of the composite function, and calculate .
The given function is:
Step 1: Break it into an inner and outer function
- Inner function:
- Outer function:
This gives the composite function .
Step 2: Chain Rule Application
By the chain rule, we calculate the derivative of the composite function:
- is the derivative of with respect to , which is:
- is the derivative of with respect to , which is:
Step 3: Combine the Results
Now, substitute back into the expression:
Thus, the derivative is:
Answer Selection
The correct choice is: D.
Do you want more details, or have any questions?
Here are five related questions to explore:
- How do we apply the chain rule to more complex functions?
- What is the derivative of a composite trigonometric function like ?
- How does the chain rule apply when there are multiple inner functions?
- Can we find the second derivative of this composite function?
- How would the derivative change if the function were ?
Tip: Always identify the inner and outer functions clearly when applying the chain rule for composite functions.
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Math Problem Analysis
Mathematical Concepts
Calculus
Composite Functions
Trigonometric Functions
Chain Rule
Formulas
y = cos(8x/5)
dy/dx = (dy/du) * (du/dx)
dy/du = -sin(u)
du/dx = 8/5
Theorems
Chain Rule
Suitable Grade Level
Grades 11-12