Math Problem Statement

deriv of 5ln(x) / 4x+3 where x = e

Solution

To find the derivative of 5ln(x)4x+3\frac{5\ln(x)}{4x + 3} and then evaluate it at x=ex = e, let's go through the steps:

Step 1: Apply the quotient rule for derivatives.

The function f(x)=u(x)v(x)f(x) = \frac{u(x)}{v(x)} where:

  • u(x)=5ln(x)u(x) = 5\ln(x)
  • v(x)=4x+3v(x) = 4x + 3

The quotient rule states: f(x)=u(x)v(x)u(x)v(x)(v(x))2f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}

Step 2: Compute u(x)u'(x) and v(x)v'(x).

  • u(x)=5ln(x)u(x) = 5\ln(x) so u(x)=5xu'(x) = \frac{5}{x}
  • v(x)=4x+3v(x) = 4x + 3 so v(x)=4v'(x) = 4

Step 3: Apply the quotient rule.

Substituting u(x)u'(x), u(x)u(x), v(x)v(x), and v(x)v'(x) into the quotient rule formula: f(x)=(5x)(4x+3)5ln(x)(4)(4x+3)2f'(x) = \frac{\left(\frac{5}{x}\right)(4x + 3) - 5\ln(x)(4)}{(4x + 3)^2}

Step 4: Simplify the derivative.

Simplify the numerator: f(x)=5(4x+3)x20ln(x)(4x+3)2f'(x) = \frac{\frac{5(4x + 3)}{x} - 20\ln(x)}{(4x + 3)^2} Now, evaluate this expression at x=ex = e.

Step 5: Evaluate at x=ex = e.

For x=ex = e:

  • ln(e)=1\ln(e) = 1
  • The expression simplifies to: f(e)=5(4e+3)e20(1)(4e+3)2f'(e) = \frac{\frac{5(4e + 3)}{e} - 20(1)}{(4e + 3)^2} Simplifying further: f(e)=5(4e+3)e20(4e+3)2=5(4e+3)e20(4e+3)2f'(e) = \frac{\frac{5(4e + 3)}{e} - 20}{(4e + 3)^2} = \frac{\frac{5(4e + 3)}{e} - 20}{(4e + 3)^2} We can now compute this expression numerically.

Let me calculate the result.The derivative of 5ln(x)4x+3\frac{5\ln(x)}{4x + 3} evaluated at x=ex = e is approximately 0.02870.0287.

Would you like further details or clarification on any step?

Here are 5 related questions to expand on this concept:

  1. How does the quotient rule apply to other types of functions?
  2. What is the general process of evaluating a derivative at a specific point?
  3. How does logarithmic differentiation help in more complex expressions?
  4. What happens when we evaluate the second derivative of this function?
  5. Can we simplify this function before differentiating to make the process easier?

Tip: When working with natural logarithms, remember that ln(e)=1\ln(e) = 1, 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)