## Calculus Problem: Derivative of a Logarithmic Hyperbolic Function### Problem Statement:Find the derivative of \( y = \ln(\tanh(x)) \).### Solution Approach:To find the derivative of \( y = \ln(\tanh(x)) \), we will use the chain rule.1. **Identify the Outer and Inner Functions**: Our function is a composition of two functions: - Outer function: \( u = \ln(u) \) - Inner function: \( u = \tanh(x) \)2. **Differentiate the Outer Function with respect to the Inner Function**: Differentiating \( \ln(u) \) with respect to \( u \), we get: \[ \frac{d}{du} \ln(u) = \frac{1}{u} \]3. **Differentiate the Inner Function with respect to \( x \)**: Differentiating \( \tanh(x) \) with respect to \( x \), we have: \[ \frac{d}{dx} \tanh(x) = 1 - \tanh^2(x) \]4. **Combine Using the Chain Rule**: By the chain rule, the derivative of \( y = \ln(\tanh(x)) \) with respect to \( x \) is: \[ \frac{dy}{dx} = \frac{d}{du} \ln(u) \cdot \frac{du}{dx} = \frac{1}{\tanh(x)} \cdot (1 - \tanh^2(x)) \]5. **Simplify the Result**: Therefore, the derivative is: \[ \frac{dy}{dx} = \frac{1 - \tanh^2(x)}{\tanh(x)} \]Thus, the derivative is:\[\boxed{\frac{dy}{dx} = \frac{1 - \tanh^2(x)}{\tanh(x)}}\]This result gives us the rate of change of the function \( y = \ln(\tanh(x)) \) with respect to \( x \).---This problem illustrates the application of the chain rule in differentiation, specifically dealing with logarithmic and hyperbolic functions. ### Problem Statement:Evaluate the integral of the hyperbolic tangent function \( \tanh(3x) \).### Mathematical Expression:\[\int \tanh(3x) \, dx\]

Name: ## Calculus Problem: Derivative of a Logarithmic Hyperbolic Function#
Uploaded: 2024-03-20T06:47:22.000Z
Channel: Bartleby
Description: ## Calculus Problem: Derivative of a Logarithmic Hyperbolic Function### Problem Statement:Find the derivative of \( y = \ln(\tanh(x)) \).### Solution Approach:To find the derivative of \( y = \ln(\tanh(x)) \), we will use the chain rule.1. **Identif