Math Problem Statement
Solution
The function to differentiate is:
To find , we will use the chain rule along with the quotient rule. Here are the steps:
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Chain Rule: First, recognize that the outer function is a square, so we will differentiate first and then multiply by the derivative of the inside.
Using the chain rule: where .
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Quotient Rule: Now, to differentiate the inside part , we use the quotient rule: where and .
Let's calculate the necessary derivatives:
Substituting these into the quotient rule: Simplifying the numerator: So the derivative becomes:
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Final derivative: Now applying the chain rule to get : Simplifying the final expression:
So, the derivative is:
Would you like a more detailed explanation of any part of this solution?
Related questions:
- How does the chain rule work in this context?
- Can you explain the quotient rule in more depth?
- What would change if the power was different, such as ?
- How can the quotient rule be generalized for more complex functions?
- What are some common mistakes to avoid when applying the quotient rule?
Tip: Always simplify the numerator and denominator before differentiating to make the process easier.
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Math Problem Analysis
Mathematical Concepts
Calculus
Differentiation
Chain Rule
Quotient Rule
Formulas
Chain Rule: d/dt (u^n) = n * u^(n-1) * du/dt
Quotient Rule: d/dt (f/g) = (f'g - fg') / g^2
Theorems
Chain Rule
Quotient Rule
Suitable Grade Level
Grades 11-12 (Advanced Calculus)
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