Math Problem Statement
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Solution
To find the derivative of the function , we will use the rules of differentiation. Here is the step-by-step solution:
Function:
Step 1: Derivative of
Using the chain rule:
Step 2: Derivative of
Using the product rule: Let and . Then: where and . Substituting these:
Final Answer:
Combining both parts:
Let me know if you want further clarification or detailed steps for a specific part!
Here are 5 questions to further your understanding:
- Can you derive independently using the chain rule?
- How does the product rule simplify differentiation in ?
- Why does have a derivative of ?
- What happens to the derivative when ?
- How does the presence of and in the function affect its rate of change?
Tip: Always check the domain of the function before differentiating, especially
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Math Problem Analysis
Mathematical Concepts
Differentiation
Logarithmic Functions
Chain Rule
Product Rule
Formulas
d/dx [√(ln x)] = 1 / (2x√(ln x))
d/dx [(1/x) log x] = (1 - log x) / x^2
Theorems
Chain Rule
Product Rule
Suitable Grade Level
Undergraduate Calculus