Math Problem Statement
deriv of 5ln(x) / 4x+3 where x = e
Solution
To find the derivative of and then evaluate it at , let's go through the steps:
Step 1: Apply the quotient rule for derivatives.
The function where:
The quotient rule states:
Step 2: Compute and .
- so
- so
Step 3: Apply the quotient rule.
Substituting , , , and into the quotient rule formula:
Step 4: Simplify the derivative.
Simplify the numerator: Now, evaluate this expression at .
Step 5: Evaluate at .
For :
- The expression simplifies to: Simplifying further: We can now compute this expression numerically.
Let me calculate the result.The derivative of evaluated at is approximately .
Would you like further details or clarification on any step?
Here are 5 related questions to expand on this concept:
- How does the quotient rule apply to other types of functions?
- What is the general process of evaluating a derivative at a specific point?
- How does logarithmic differentiation help in more complex expressions?
- What happens when we evaluate the second derivative of this function?
- Can we simplify this function before differentiating to make the process easier?
Tip: When working with natural logarithms, remember that , which can often simplify your calculations!
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Math Problem Analysis
Mathematical Concepts
Differentiation
Logarithmic Functions
Quotient Rule
Formulas
Quotient Rule: f'(x) = (u'(x)v(x) - u(x)v'(x)) / (v(x))^2
Derivative of ln(x): d/dx[ln(x)] = 1/x
Theorems
Quotient Rule for Derivatives
Suitable Grade Level
Grades 11-12 (Calculus)