Problem 17 - Thomas Calculus

Question

In Exercises \(13-24,\) find the derivative of \(y\) with respect to the appropriate variable. $$y=\ln (\sinh z)$$

Step-by-Step Solution

Answer

The derivative is \( \frac{dy}{dz} = \coth z \).

Step 1: Understand the Function

The function given is \( y = \ln(\sinh z) \). It represents the natural logarithm of the hyperbolic sine function. Our goal is to find the derivative \( \frac{dy}{dz} \).

Step 2: Know the Basic Derivative Rules

To solve this problem, we will use the chain rule for derivatives combined with the derivatives of specific functions: the derivative of \( \ln(u) \) is \( \frac{1}{u} \cdot \frac{du}{dz} \) and the derivative of \( \sinh z \) is \( \cosh z \).

Step 3: Apply the Chain Rule

First, identify the inner function as \( u = \sinh z \). The derivative of \( y = \ln(u) \) with respect to \( u \) is \( \frac{dy}{du} = \frac{1}{u} \).

Step 4: Differentiate the Inner Function

Differentiate the inner function \( u = \sinh z \) with respect to \( z \). The derivative is \( \frac{du}{dz} = \cosh z \).

Step 5: Combine the Derivatives Using the Chain Rule

Substitute the derivative of the inner function into the formula from Step 3: \( \frac{dy}{dz} = \frac{1}{\sinh z} \cdot \cosh z \).

Step 6: Simplify the Expression

The derivative simplifies to \( \frac{dy}{dz} = \coth z \), since \( \coth z = \frac{\cosh z}{\sinh z} \).

Key Concepts

Chain Rule

The chain rule is an essential technique in calculus for finding the derivative of composite functions. When you have a function nested inside another function, consider using the chain rule. It allows you to differentiate complex expressions by breaking them down into simpler parts.

• First, identify the inner and outer functions. In our example, the inner function is \( u = \sinh z \) and the outer function is \( y = \ln(u) \).
• Derive the outer function with respect to the inner function: \( \frac{dy}{du} = \frac{1}{u} \).
• Next, differentiate the inner function in terms of its variable: \( \frac{du}{dz} = \cosh z \).
• The chain rule states that the derivative of the composite function is the product of these derivatives: \( \frac{dy}{dz} = \frac{dy}{du} \cdot \frac{du}{dz} \).

The chain rule is a powerful method. It allows you to solve complex derivatives systematically by following these steps.

Hyperbolic Functions

Hyperbolic functions resemble trigonometric functions and are often used in calculus to solve problems involving certain kinds of differential equations. They are based on hyperbolas, unlike trig functions, which are based on circles.

• The hyperbolic sine function, \( \sinh z \), is defined as \( \sinh z = \frac{e^z - e^{-z}}{2} \).
• The derivative of \( \sinh z \) is the hyperbolic cosine function, \( \cosh z \), which is \( \cosh z = \frac{e^z + e^{-z}}{2} \).
• Hyperbolic functions have identities similar to trigonometric ones, such as \( \cosh^2 z - \sinh^2 z = 1 \).

These functions have unique properties and applications, especially in hyperbolic geometry and many physics problems.

Natural Logarithm

The natural logarithm, often written as \( \ln \, x \), is the inverse of the exponential function \( e^x \). It is commonly used in calculus because of its special properties, which make differentiation straightforward.

• The derivative of \( \ln x \) is \( \frac{1}{x} \), provided that \( x > 0 \).
• The natural logarithm is base \( e \), where \( e \approx 2.71828 \), making it natural for calculations involving continuous growth models, like populations or radioactive decay.
• In the exercise solution, understanding the property of \( \ln(x) \) being \( \frac{1}{x} \) was pivotal in applying the chain rule: leading to the simplification of the derivative.

The natural logarithm is crucial in mathematics for modeling real-world problems due to its constant \( e \) and its ease of integration and differentiation.