Math Problem Statement
find the derivative
Solution
The given function is . To find the derivative, we'll use the product rule along with the rules of differentiation.
Step-by-Step Solution
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Rewrite the function:
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Apply the product rule: The product rule states that: Here, and .
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Differentiate each term:
- gives ,
- gives (using the power rule).
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Substitute into the product rule:
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Simplify the expression:
- First term: ,
- Second term: .
Add the terms:
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Combine like terms:
Final Answer:
Would you like further details or examples on this type of differentiation?
Questions to explore further:
- How does the product rule work for more complex functions?
- Can this derivative be applied to real-world problems involving roots?
- What happens if in this context (domain issues)?
- How does this derivative change when combined with other operations?
- What are other scenarios where roots and powers are involved in differentiation?
Tip: Always simplify roots and exponents to their fractional forms for easier differentiation.
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Math Problem Analysis
Mathematical Concepts
Differentiation
Exponents and Roots
Product Rule
Formulas
Product Rule: d(uv)/dx = u'v + uv'
Power Rule: d(x^n)/dx = nx^(n-1)
Theorems
Product Rule of Differentiation
Power Rule for Exponents
Suitable Grade Level
Grades 11-12