**Problem:** Find the derivative of the function.\[ h(x) = \ln{[(8x + 6)(2x + 1)]} \]**Solution:** To find the derivative of the function \( h(x) \), we can use the chain rule and properties of logarithms. Start by using the logarithmic property that allows us to separate the products inside the logarithm:\[ h(x) = \ln{(8x + 6)} + \ln{(2x + 1)} \]Next, we differentiate each term independently. Using the chain rule:For \( \ln{(8x + 6)} \):\[ \frac{d}{dx} \ln{(8x + 6)} = \frac{1}{8x + 6} \cdot 8 = \frac{8}{8x + 6} \]For \( \ln{(2x + 1)} \):\[ \frac{d}{dx} \ln{(2x + 1)} = \frac{1}{2x + 1} \cdot 2 = \frac{2}{2x + 1} \]Then, we sum the derivatives:\[ h'(x) = \frac{8}{8x + 6} + \frac{2}{2x + 1} \]This is the derivative of the given function \( h(x) \). Title: Calculating the Derivative of a Logarithmic FunctionObjective: Learn how to find the derivative of a given logarithmic function.Problem Statement:Find the derivative of the following function.Function:\[ f(x) = \log_{4}(8x) \]Steps to Solution:1. **Identify the Function Structure**: The given function is a logarithmic function with a variable base. 2. **Apply the Chain Rule**: The derivative of \( \log_b(u) \) with respect to \( x \) is given by: \[ \frac{d}{dx} [ \log_b(u) ] = \frac{1}{u \ln(b)} \cdot \frac{du}{dx} \] where \( u = 8x \) and \( b = 4 \).3. **Calculate the Inner Function Derivative** First, find the derivative of the inner function \( u = 8x \): \[ \frac{du}{dx} = 8 \]4. **Compute the Derivative of the Logarithmic Function**: Substitute \( u = 8x \), \( b = 4 \), and \( \frac{du}{dx} = 8 \) into the chain rule formula: \[ \frac{d}{dx} [ \log_{4}(8x) ] = \frac{1}{8x \ln(4)} \cdot 8 \] 5. **Simplify the Expression**: Simplify the resulting expression: \[ f'(x) = \frac{8}{8x \ln(4)} = \frac{1}{x \ln(4)} \]Final Answer:\[f'(x) = \frac{1}{x \ln(4)}\]Conclusion:This calculation shows that the derivative of the function \( f(x) = \log_{4}(8x) \) is \( \frac{1}{x \ln(4)} \). Understanding and applying the chain rule is essential in finding the derivative of composite functions, especially when dealing with logarithms with variable bases.Illustration:- This problem does not contain diagrams or graphs that need detailed explanations.

Name: **Problem:** Find the derivative of the function.\[ h(x) = \ln{[(8x +
Uploaded: 2024-03-20T06:47:22.000Z
Channel: Bartleby
Description: **Problem:** Find the derivative of the function.\[ h(x) = \ln{[(8x + 6)(2x + 1)]} \]**Solution:** To find the derivative of the function \( h(x) \), we can use the chain rule and properties of logarithms. Start by using the logarithmic property tha