Problem 21 - Thomas Calculus

Question

In Exercises \(13-24,\) find the derivative of \(y\) with respect to the appropriate variable. $$y=\ln \cosh v-\frac{1}{2} \tanh ^{2} v$$

Step-by-Step Solution

Answer

The derivative is \( \tanh v - \tanh v \cdot \text{sech}^2 v \).

Step 1: Differentiate the first term

We need to find the derivative of \( \ln \cosh v \) with respect to \( v \). Using the chain rule, the derivative of \( \ln u \) is \( \frac{1}{u} \cdot \frac{du}{dv} \), where \( u = \cosh v \). Thus, the derivative of the first term is \(\frac{1}{\cosh v} \cdot \sinh v = \tanh v.\)

Step 2: Differentiate the second term

We now differentiate \( -\frac{1}{2} \tanh^2 v \) with respect to \( v \). Using the chain rule and the power rule, the derivative of \( \tanh^2 v \) is \( 2\tanh v \cdot \text{sech}^2 v \). Thus, the derivative of the second term is \( -\frac{1}{2} \times 2 \tanh v \cdot \text{sech}^2 v = -\tanh v \cdot \text{sech}^2 v \).

Step 3: Combine the derivatives

Add the derivatives from Step 1 and Step 2 to find the derivative of \( y \) with respect to \( v \): \( \tanh v - \tanh v \cdot \text{sech}^2 v \).

Key Concepts

Chain Rule in Calculus

When finding derivatives, the chain rule is a fundamental tool. It's particularly useful when dealing with compositions of functions. In simple terms, if you have a function within another function, like in our example with \( \ln \cosh v \), the chain rule aids in differentiating this nested function. Here’s how it works:

• Suppose you have a function \( f(g(x)) \).
• The chain rule states that the derivative is \( f'(g(x)) \cdot g'(x) \).

This means you differentiate the outer function (\( f \)) and multiply it by the derivative of the inner function (\( g \)). In the example provided, the chain rule helps differentiate \( \ln \cosh v \) by first taking the derivative of \( \ln u \), which is \( \frac{1}{u} \), and then multiplying it with the derivative of \( \cosh v \), which is \( \sinh v \).
This becomes a systematic approach when faced with more complex functions, ensuring accuracy in computing derivatives.

Understanding Hyperbolic Functions

Hyperbolic functions often arise in calculus problems and have similar names and properties to trigonometric functions, yet they differ quite a bit. The main hyperbolic functions are \( \sinh x \), \( \cosh x \), and \( \tanh x \). Let's break these down:

• \( \sinh v = \frac{e^v - e^{-v}}{2} \)
• \( \cosh v = \frac{e^v + e^{-v}}{2} \)
• \( \tanh v = \frac{\sinh v}{\cosh v} \)

These functions model hyperbolic curves, similar to how sine and cosine model circular curves. They appear in various contexts in mathematics, physics, and engineering applications.
Knowing the derivatives of these hyperbolic functions also helps in solving calculus problems. For instance:

• The derivative of \( \sinh v \) is \( \cosh v \)
• The derivative of \( \cosh v \) is \( \sinh v \)

Understanding these functions and their derivatives is crucial when encountering calculus problems involving hyperbolic expressions.

Differentiation Steps Simplified

Differentiation involves finding how a function changes as its input changes, defined as the derivative. Finding derivatives can be broken down into simple steps:

• Identify the function components. For \( y=\ln \cosh v - \frac{1}{2} \tanh^2 v \), recognize it as two separate functions.
• Apply rules like chain rule, product rule, or power rule as necessary. Here, both terms were handled using the chain and power rules.
• Compute each term's derivative separately. Step 1 in the solution took \( \ln \cosh v \), and with the chain rule, found \( \tanh v \). Step 2 dealt with \( -\frac{1}{2} \tanh^2 v \) using the chain and power rule to deduce \( -\tanh v \cdot \text{sech}^2 v \).
• Combine the results. Finally, adding them gives the full derivative.

Following these steps ensures all aspects of the function are adequately differentiated, making the problem more manageable and systematic.

Power Rule for Derivatives

The power rule is one of the simplest and most commonly used rules in differentiation. It states that for any function \( x^n \), where \( n \) is a real number, the derivative is \( nx^{n-1} \). This rule makes differentiating polynomial functions straightforward.
In the context of the given exercise, the power rule was particularly useful in differentiating \( \tanh^2 v \). Here’s how:

• Recognize \( \tanh^2 v \) as a composition of a function. First, the derivative of \( v^2 \) is \( 2v \).
• Apply the power rule \( 2x^{2-1} \), giving \( 2x \).
• Combine with the chain rule, considering the derivative of the inside function \( \tanh v \) itself (which is \( \text{sech}^2 v \)).

The power rule simplifies many components of differentiation, providing efficient results when used alongside other rules. Knowing when and how to apply it can greatly speed up the process of finding derivatives.