Math Problem Statement
Differentiate the function. If possible, first use properties of logarithms to simplify the given function. f(x)equalsln left parenthesis StartFraction x cubed minus 4 Over x EndFraction right parenthesis
Solution
To differentiate the function , we can first simplify it using the properties of logarithms. Specifically, we use the property:
Step 1: Simplify using logarithmic properties
Apply the property to the given function:
Step 2: Differentiate the simplified function
Now we differentiate each term separately:
- For the term , we apply the chain rule:
- For the term , the derivative is:
Step 3: Combine the results
The derivative of is:
This is the derivative of the given function.
Would you like further clarification or more details on this? Here are some related questions:
- How do logarithmic properties help in differentiation?
- Can we differentiate without using logarithmic simplification?
- What is the chain rule in calculus?
- How do you simplify complex fractions after differentiation?
- Can this method be applied to other types of logarithmic functions?
Tip: Simplifying a function before differentiation often leads to easier and faster results.
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Math Problem Analysis
Mathematical Concepts
Differentiation
Logarithmic Functions
Chain Rule
Formulas
ln(a/b) = ln(a) - ln(b)
Derivative of ln(x) = 1/x
Chain Rule: d/dx[f(g(x))] = f'(g(x)) * g'(x)
Theorems
Logarithmic Differentiation
Chain Rule
Suitable Grade Level
Grades 11-12, Early College
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