Math Problem Statement
Solution
To determine when , we use differentiation rules and evaluate at .
1. Function Setup
The given function is:
This is a quotient of two functions, so we need the quotient rule for derivatives.
2. Derivative Using Quotient Rule
The quotient rule states:
Here:
Step 1: Differentiate
We simplify first:
Now differentiate:
Step 2: Differentiate
The denominator . Using the chain rule:
3. Apply Quotient Rule
Substitute into the quotient rule:
At , we calculate the following:
Step 1: Compute and
Step 2: Compute and
Step 3: Substitute into the Quotient Rule
Now:
Substitute the values:
- , , ,
Simplify:
4. Final Answer
The correct answer is:
Do you want further details or explanations on any step?
Related Questions:
- What is the derivative of a product of two functions (product rule)?
- How do you differentiate a square root function using the chain rule?
- How does the quotient rule work for derivatives?
- What happens if a numerator or denominator equals zero at a specific point?
- How do you simplify derivatives involving radicals?
Tip:
Always simplify expressions as much as possible before differentiating to make calculations easier.
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Math Problem Analysis
Mathematical Concepts
Differentiation
Quotient Rule
Chain Rule
Formulas
Quotient Rule: \( \left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2} \)
Chain Rule: \( \frac{d}{dx} [g(h(x))] = g'(h(x)) \cdot h'(x) \)
Theorems
Quotient Rule for Derivatives
Chain Rule for Derivatives
Suitable Grade Level
Undergraduate Calculus, AP Calculus AB/BC