Math Problem Statement
Solution
The image contains exercises related to derivatives of natural logarithmic functions. Let’s work through a few examples step-by-step while using the properties of logarithms to simplify first.
Example Solutions:
41.
Using the property of logarithms: , The derivative is:
42.
Using the derivative formula , where and ,
43.
Using the property , The derivative is:
44.
Using the derivative formula , where and ,
45.
Using the chain rule, let so . Then, where . Substituting:
Would you like me to proceed with solving more examples or explain any of the steps in further detail?
Questions to Explore:
- What are the general rules for logarithmic differentiation?
- How can the chain rule and logarithmic properties simplify differentiation problems?
- What is the role of implicit differentiation in logarithmic derivatives?
- How do we handle derivatives of logarithms involving nested functions?
- Can logarithmic differentiation simplify differentiation of complex rational functions?
Tip:
Always simplify logarithmic expressions using log properties before differentiating to make calculations easier!
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Math Problem Analysis
Mathematical Concepts
Differentiation
Natural Logarithms
Logarithmic Properties
Chain Rule
Formulas
d/dx [ln(u)] = u'/u
ln(ab) = ln(a) + ln(b)
ln(a^n) = nln(a)
ln(a/b) = ln(a) - ln(b)
Theorems
Rules of Differentiation
Logarithmic Differentiation
Suitable Grade Level
Grades 11-12 (High School Calculus)
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