### Derivative of Hyperbolic Functions**Problem Statement:**Find the derivative of the given function:\[ \ln(\text{sech}(x) + \tanh(x)) \]**Multiple Choice Answers:**1. \[ \frac{\text{sech } x + \tanh x}{\sinh x + 1} \]2. \[ \frac{\text{sech } x - \tanh x}{\sinh x - 1} \]3. \[ \frac{\text{sech } x}{\sinh x + 1} \]4. \[ \frac{\text{sech } x - \tanh x}{\sinh x + 1} \] (Highlighted as the correct answer)### Explanation:The given problem is asking for the derivative of the natural logarithm of the sum of the hyperbolic secant and hyperbolic tangent functions. To solve this problem, we can use the chain rule and properties of derivatives for hyperbolic functions.### Steps to Derive:1. **Identify the outer function and the inner function:** - Outer function: \(\ln(u)\) where \(u = \text{sech}(x) + \tanh(x)\) 2. **Apply the chain rule:** \[ \frac{d}{dx}[\ln(u)] = \frac{1}{u} \cdot \frac{du}{dx} \]3. **Differentiate the inner function \(u\):** \[ u = \text{sech}(x) + \tanh(x) \] The derivative of \( \text{sech}(x) \) is \( -\text{sech}(x)\tanh(x) \) and the derivative of \( \tanh(x) \) is \( \text{sech}^2(x) \). \[ \frac{du}{dx} = -\text{sech}(x)\tanh(x) + \text{sech}^2(x) \]4. **Combine the derivatives:** \[ \frac{d}{dx}[\ln(\text{sech}(x) + \tanh(x))] = \frac{1}{\text{sech}(x) +

Name: ### Derivative of Hyperbolic Functions**Problem Statement:**Find the
Uploaded: 2024-03-20T06:47:22.000Z
Channel: Bartleby
Description: ### Derivative of Hyperbolic Functions**Problem Statement:**Find the derivative of the given function:\[ \ln(\text{sech}(x) + \tanh(x)) \]**Multiple Choice Answers:**1. \[ \frac{\text{sech } x + \tanh x}{\sinh x + 1} \]2. \[ \frac{\text{sech } x - \