Need help with homework Determine g’(x) **Problem Statement:**Determine \( g'(x) \) if \( g(x) = \ln \left( \frac{x^2 + 5x + 8}{x^3 + 5x + 8} \right) \).**Solution Overview:**To find the derivative \( g'(x) \), we will use the chain rule and the properties of logarithmic differentiation. The function \( g(x) \) is a natural logarithm of a fraction, specifically \( \ln(u(x)) \) where:\[ u(x) = \frac{x^2 + 5x + 8}{x^3 + 5x + 8} \]### Steps:1. **Differentiate the logarithmic function:** Using the derivative of the natural logarithm, \((\ln(u))' = \frac{1}{u} \cdot u'\).2. **Apply the quotient rule to find \( u'(x) \):** For a function \( \frac{f(x)}{h(x)} \), the quotient rule states: \[ \left(\frac{f}{h}\right)' = \frac{f'h - fh'}{h^2} \]3. **Determine \( f(x) = x^2 + 5x + 8 \) and \( h(x) = x^3 + 5x + 8 \):** - \( f'(x) = 2x + 5 \) - \( h'(x) = 3x^2 + 5 \)4. **Apply the quotient rule:** \[ u'(x) = \frac{(2x + 5)(x^3 + 5x + 8) - (x^2 + 5x + 8)(3x^2 + 5)}{(x^3 + 5x + 8)^2} \]5. **Plug \( u \) and \( u' \) into the derivative of the logarithmic function:** \[ g'(x) = \frac{1}{u(x)} \cdot u'(x) \] Simplify this expression for the final result. Below is the transcription of the mathematical derivatives as shown in the image. Equation 1:\[ g'(x) = \frac{2x - 5}{x^2 + 5x + 8} - \frac{3x^2 - 5}{x^3 + 5x + 8} \]Equation 2:\[ g'(x) = \frac{x^2 + 5x + 8}{2x + 5} - \frac{x^3 + 5x + 8}{3x^2 + 5} \]Equation 3:\[ g'(x) = \frac{2x + 5}{x^2 + 5x + 8} + \frac{3x^2 + 5}{x^3 + 5x + 8} \]There are no graphs or diagrams to explain in this image. The image consists solely of these mathematical expressions.

Name: Need help with homework Determine g’(x) **Problem Statement:**Determin
Uploaded: 2024-03-20T06:47:22.000Z
Channel: Bartleby
Description: Need help with homework Determine g’(x) **Problem Statement:**Determine \( g'(x) \) if \( g(x) = \ln \left( \frac{x^2 + 5x + 8}{x^3 + 5x + 8} \right) \).**Solution Overview:**To find the derivative \( g'(x) \), we will use the chain rule and the prop