Math Problem Statement
Solution
We are asked to find the derivative of the following function:
Let's break it down step by step:
Step 1: Simplify the expression inside the logarithm
We know that , where is a function of . Applying this rule, we can rewrite the expression as:
Step 2: Use the logarithmic property
This allows us to separate the terms:
Step 3: Differentiate using the chain rule
Now, we differentiate each part with respect to . Let's handle each term separately:
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For , the derivative is:
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For , the derivative is:
Step 4: Combine the results
Now, putting it all together:
Step 5: Factor and simplify
We can factor out the common term :
Step 6: Combine the fractions
We can combine the two fractions under a common denominator:
Simplifying the numerator:
Final Answer:
This is the derivative of the given function.
Would you like further details or have any questions?
5 Related Questions:
- How would the result change if the logarithmic function was natural log squared?
- Can we find the second derivative of this function?
- What is the domain of the original function?
- How does this process change for logarithms with a different base (e.g., log base 10)?
- How does simplifying the function before differentiation help in solving the derivative?
Tip:
When differentiating complex expressions, breaking them down using logarithmic properties and simplifying before applying the chain rule can save a lot of time and reduce errors.
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Math Problem Analysis
Mathematical Concepts
Logarithmic Differentiation
Chain Rule
Quotient Rule
Simplification
Formulas
d/dx [ln(u)] = (1/u) * (du/dx)
d/dx [sqrt(u)] = (1/2sqrt(u)) * (du/dx)
ln(a/b) = ln(a) - ln(b)
Theorems
Logarithmic Differentiation
Chain Rule
Suitable Grade Level
Undergraduate Calculus or Advanced High School Calculus