Problem 394 - Calculus Volume 1

Question

For the following exercises, find the derivatives of the given functions and graph along with the function to ensure your answer is correct. $$\ln (\operatorname{sech}(x)+\tanh (x))$$

Step-by-Step Solution

Answer

The derivative is \( f'(x) = \frac{\operatorname{sech}^2(x) - \operatorname{sech}(x) \tanh(x)}{\operatorname{sech}(x) + \tanh(x)} \).

Step 1: Understand the function

The function given is \( f(x) = \ln (\operatorname{sech}(x) + \tanh(x)) \). This combines the hyperbolic functions \( \operatorname{sech}(x) \) and \( \tanh(x) \) inside the natural log function \( \ln \). Our goal is to find \( f'(x) \).

Step 2: Use the chain rule and implicit differentiation

The derivative of a natural log function \( \ln(u) \) is \( \frac{1}{u} \cdot \frac{du}{dx} \). In this exercise, \( u = \operatorname{sech}(x) + \tanh(x) \).Identify derivatives of individual components: - Derivative of \( \operatorname{sech}(x) \) is \(-\operatorname{sech}(x) \tanh(x) \).- Derivative of \( \tanh(x) \) is \( \operatorname{sech}^2(x) \).

Step 3: Apply the derivatives to the function

Apply the derivatives to get: \[ f'(x) = \frac{1}{\operatorname{sech}(x) + \tanh(x)} \cdot \left(-\operatorname{sech}(x) \tanh(x) + \operatorname{sech}^2(x)\right) \]

Step 4: Simplify the expression

Simplify the expression substituting back the hyperbolic functions' identities: \[ f'(x) = \frac{\operatorname{sech}^2(x) - \operatorname{sech}(x) \tanh(x)}{\operatorname{sech}(x) + \tanh(x)} \]This step involves algebraic simplification.

Step 5: Graph the function and its derivative

Plot the function \( f(x) = \ln (\operatorname{sech}(x) + \tanh(x)) \) and its derivative \( f'(x) \). Check if the derivative graph aligns with the slope of \( f(x) \) at various points to validate the correctness. If the derivative graph looks like the tangent lines to \( f(x) \), it confirms correctness.

Key Concepts

Chain Rule

The chain rule is a fundamental concept in calculus used to find the derivative of composite functions. A composite function is essentially one function inside another, like our example here: a natural logarithm function involving hyperbolic functions.

• The chain rule states that if you have a function within another function, such as \( f(g(x)) \), the derivative \( f'(x) \) can be found by taking the derivative of the outside function evaluated at the inside function \( g(x) \), and multiplying it by the derivative of the inside function \( g'(x) \).
• In symbolic terms, this is represented as \( \frac{d}{dx}f(g(x)) = f'(g(x)) \cdot g'(x) \).

So in our problem, you calculate the derivative of the outer natural logarithm function \( \ln(u) \) and multiply it by the derivative of the inside hyperbolic sum \( \operatorname{sech}(x) + \tanh(x) \). This is why the derivative starts as \( \frac{1}{u} \cdot \frac{du}{dx} \). Understanding the chain rule is critical when handling layers of functions, as it ensures you differentiate correctly at each step.

Hyperbolic Functions

Hyperbolic functions are analogs of trigonometric functions but for a hyperbola rather than a circle. They include hyperbolic sine \( \sinh \), hyperbolic cosine \( \cosh \), hyperbolic tangent \( \tanh \), and hyperbolic secant \( \operatorname{sech} \).

• Hyperbolic tangent, \( \tanh(x) \), is defined as \( \frac{\sinh(x)}{\cosh(x)} \).
• Hyperbolic secant, \( \operatorname{sech}(x) \), is defined as \( \frac{1}{\cosh(x)} \).
• The derivatives of hyperbolic functions, much like their trigonometric counterparts, follow specific patterns: \( \operatorname{sech}'(x) = -\operatorname{sech}(x) \tanh(x) \) and \( \tanh'(x) = \operatorname{sech}^2(x) \).

These functions and their derivatives play a crucial role in solving our problem. By knowing these derivatives, you can apply them as needed within the chain rule framework to solve the exercise.

Natural Logarithm

The natural logarithm function, denoted as \( \ln(x) \), is a pivotal tool in calculus, characterized by the base \( e \), where \( e \approx 2.718 \. \) The natural logarithm serves several purposes including simplifying multiplication to addition, and its derivative properties are especially important in calculus.

• The derivative of \( \ln(x) \) is \( \frac{1}{x} \), meaning it provides a straightforward result when differentiating logs.
• When combined with other functions, as in our case with hyperbolic functions inside the log, it often requires the use of the chain rule due to the composition of functions.
• The natural log is widely used because of this property and because it simplifies certain calculations, transforming multiplicative processes into additive ones.

In our problem, the natural logarithm of the sum of \( \operatorname{sech}(x) \) and \( \tanh(x) \) requires you to apply these derivative principles, using the chain rule to evaluate the full derivative carefully. By breaking the function into parts and using the properties of \( \ln(x) \), you find the derivative efficiently.