Math Problem Statement
Solution
We are tasked with finding the derivative of the function:
We will use the chain rule to differentiate this function.
Step-by-step solution:
- Outer function: The outer function is , where . The derivative of with respect to is:
Thus, applying the chain rule:
-
Inner function: Now we need to find where .
- Use the quotient rule for the derivative of .
- Let and . The quotient rule states:
- (the derivative of ),
- , so (the derivative of a square root function).
Now applying the quotient rule:
- Simplifying the expression:
The denominator simplifies to , and the numerator becomes:
Thus, the derivative simplifies to:
- Final derivative: Now substitute back into the original expression for :
Answer:
Let me know if you want further details or explanations.
Here are 5 related questions for deeper understanding:
- How does the chain rule apply in this context, and why is it necessary?
- What other functions, besides sine, commonly require the chain rule in their derivatives?
- How would the derivative change if the function involved a cosine instead of a sine?
- Can you derive the derivative of and compare the process?
- How does the quotient rule simplify complex fractions in calculus?
Tip: The quotient rule is especially useful when differentiating expressions involving two functions divided by each other, but it’s important to apply it carefully to avoid algebraic errors.
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Math Problem Analysis
Mathematical Concepts
Calculus
Chain Rule
Quotient Rule
Trigonometry
Formulas
Chain Rule: d/dt[f(g(t))] = f'(g(t)) * g'(t)
Quotient Rule: d/dt[f(t)/g(t)] = (f'(t)g(t) - f(t)g'(t)) / [g(t)]^2
Derivative of sin(u) = cos(u)
Theorems
Chain Rule
Quotient Rule
Suitable Grade Level
Grades 11-12 / Early College
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