Math Problem Statement
Solution
We are tasked with finding the derivative , where the function is given by:
We will use the quotient rule to differentiate this function. The quotient rule states that if , then the derivative is:
Here, and .
Step 1: Differentiate
We have: Differentiating term by term:
Step 2: Differentiate
We have: Using the chain rule, we get: Using the double angle identity, , we simplify:
Step 3: Apply the Quotient Rule
Now, applying the quotient rule:
Step 4: Evaluate at
We now substitute into the expressions.
For :
For :
For :
Thus, the derivative at is:
So, the correct answer is .
Would you like more details on any of the steps?
Here are 5 related questions:
- How do you use the chain rule in trigonometric functions?
- What is the significance of the quotient rule in calculus?
- How do trigonometric identities simplify derivative calculations?
- How would this calculation change if the function were altered slightly, like replaced by ?
- How can you interpret the result geometrically at ?
Tip: Always simplify the derivative expression as much as possible before plugging in specific values to avoid calculation errors.
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Math Problem Analysis
Mathematical Concepts
Calculus
Derivatives
Trigonometric Functions
Formulas
Quotient rule: f'(x) = (g'(x)h(x) - g(x)h'(x)) / (h(x))^2
Chain rule for derivatives
Double angle identity: 2sin(2x)cos(2x) = sin(4x)
Theorems
Quotient Rule
Chain Rule
Trigonometric Identities
Suitable Grade Level
College level calculus
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