Please help me this. First time learner. :-) ---**Topic: Logarithmic Differentiation**Use logarithmic differentiation to find \(\frac{dy}{dx}\) for the function:\[y = \frac{12xe^x}{\sqrt{3x + 2}}\]---**Explanation:**Logarithmic differentiation is a technique used to differentiate functions in the form of products or quotients. It simplifies the differentiation process by taking the natural logarithm of both sides, allowing for easier application of the chain rule and product/quotient rules.1. Take the natural logarithm of both sides: \(\ln(y) = \ln\left(\frac{12xe^x}{\sqrt{3x + 2}}\right)\).2. Apply logarithmic properties to simplify: \(\ln(y) = \ln(12) + \ln(x) + \ln(e^x) - \ln(\sqrt{3x + 2})\).3. Differentiate implicitly with respect to \(x\): \(\frac{1}{y} \frac{dy}{dx} = \frac{1}{x} + 1 - \frac{1}{2(3x + 2)} \cdot 3\).4. Solve for \(\frac{dy}{dx}\): \(\frac{dy}{dx} = y \left(\frac{1}{x} + 1 - \frac{3}{2(3x + 2)}\right)\).5. Substitute back for \(y\) using the original function: \(\frac{dy}{dx} = \frac{12xe^x}{\sqrt{3x + 2}} \left(\frac{1}{x} + 1 - \frac{3}{2(3x + 2)}\right)\).This method is especially useful for complex functions like products and quotients of exponentials, polynomials, and roots.---

Name: Please help me this. First time learner. :-) ---**Topic: Logarithmic D
Uploaded: 2024-03-20T07:06:41.000Z
Channel: Bartleby
Description: Please help me this. First time learner. :-) ---**Topic: Logarithmic Differentiation**Use logarithmic differentiation to find \(\frac{dy}{dx}\) for the function:\[y = \frac{12xe^x}{\sqrt{3x + 2}}\]---**Explanation:**Logarithmic differentiation is a t