Math Problem Statement
Solution
Let's work through each of the problems you've uploaded:
Problem 6: Differentiate the function
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Outer derivative (ln): The derivative of with respect to is .
- Here, , so the first part of the derivative is .
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Inner derivative (ln(2s)): The derivative of is , using the chain rule.
Thus, the derivative of is:
Problem 7: Find and for
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First derivative :
- This is a product rule problem. The product rule is .
- Let and .
- Derivative of is .
- Derivative of is (since ).
Applying the product rule:
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Second derivative :
- Use the product rule again for and combine with the derivative of .
The second derivative is:
Problem 8: Differentiate and find the domain.
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Quotient rule for derivative: The quotient rule is .
- , so .
- , so (derivative of ).
Applying the quotient rule:
= \frac{1 - \ln(x - 7) + \frac{x}{x - 7}}{(1 - \ln(x - 7))^2}$$ -
Domain:
- The domain is restricted by the natural logarithm, so , meaning .
- Additionally, , so , or , which gives .
Therefore, the domain is .
Problem 9: Use logarithmic differentiation to find the derivative of
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Take the natural logarithm of both sides:
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Differentiate implicitly:
Therefore, multiplying both sides by , we get:
Let me know if you'd like more details or have any questions! Here are five related questions:
- How is the chain rule applied to logarithmic functions?
- Can you find the second derivative of ?
- What are the conditions for applying the quotient rule?
- Can you derive the product rule for a more complex function, like ?
- How does logarithmic differentiation simplify the process for differentiating products of functions?
Tip: Always double-check the domain of logarithmic functions to ensure the argument is positive!
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Math Problem Analysis
Mathematical Concepts
Differentiation
Logarithmic Differentiation
Quotient Rule
Product Rule
Chain Rule
Implicit Differentiation
Formulas
Derivative of ln(u): d/dx [ln(u)] = 1/u * du/dx
Product Rule: (uv)' = u'v + uv'
Quotient Rule: (u/v)' = (u'v - uv') / v^2
Derivative of ln(x): d/dx [ln(x)] = 1/x
Implicit Differentiation
Theorems
Chain Rule
Product Rule
Quotient Rule
Logarithmic Differentiation
Suitable Grade Level
Grade 12, College-level Calculus
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