## Derivative of a Logarithmic FunctionIn this problem, we are asked to find the derivative of the natural logarithm function:\[ \ln \left( \frac{(5x+8)^3}{x^6+10} \right) \]### Steps to Solve1. **Simplify the Logarithmic Expression:** Using the properties of logarithms, we can simplify the given expression: \[ \ln \left( \frac{(5x+8)^3}{x^6+10} \right) = \ln (5x+8)^3 - \ln (x^6+10) \] According to the power rule of logarithms \( \ln (a^b) = b \ln a \): \[ = 3 \ln (5x+8) - \ln (x^6+10) \]2. **Differentiate the Simplified Expression:** Now, we can find the derivative of this simplified expression: \[ \frac{d}{dx} \left[ 3 \ln (5x+8) - \ln (x^6+10) \right] \] Applying the chain rule and the derivative of the natural logarithm function \( \frac{d}{dx} [\ln u] = \frac{1}{u} \cdot \frac{du}{dx} \): \[ = 3 \cdot \frac{1}{5x+8} \cdot \frac{d}{dx} [5x+8] - \frac{1}{x^6+10} \cdot \frac{d}{dx} [x^6+10] \]3. **Evaluate the Derivatives of the Inside Functions:** \[ \frac{d}{dx} [5x+8] = 5 \] \[ \frac{d}{dx} [x^6+10] = 6x^5 \]4. **Combine the Results:** Substituting these derivatives back into the expression: \[ = 3 \cdot \frac{5}{5x+8} - \frac{6x^5}{x^6+10} \] Simplify the terms: \[ = \frac{15}{5x+8} - \frac

Name: ## Derivative of a Logarithmic FunctionIn this problem, we are asked
Uploaded: 2024-03-20T06:47:22.000Z
Channel: Bartleby
Description: ## Derivative of a Logarithmic FunctionIn this problem, we are asked to find the derivative of the natural logarithm function:\[ \ln \left( \frac{(5x+8)^3}{x^6+10} \right) \]### Steps to Solve1. **Simplify the Logarithmic Expression:** Using the p