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 of \(y = \ln(\sinh z)\) with respect to \(z\) is \(\coth z\).

Step 1: Identify the Function for Differentiation

The function given is \(y = \ln (\sinh z)\). Here, \(y\) is expressed as a natural logarithm of the hyperbolic sine function. The task is to find \(\frac{dy}{dz}\), the derivative of \(y\) with respect to \(z\).

Step 2: Apply the Chain Rule

To differentiate \(y = \ln (\sinh z)\) with respect to \(z\), we need to use the chain rule, which states: if \(y = f(g(z))\), then \(\frac{dy}{dz} = f'(g(z)) \, g'(z)\). Here, \(f(u) = \ln u\) and \(g(z) = \sinh z\).

Step 3: Differentiate the Outer Function

The derivative of \(f(u) = \ln u\) with respect to \(u\) is \(f'(u) = \frac{1}{u}\). Thus, the derivative of \(\ln (\sinh z)\) with respect to \(\sinh z\) is \(\frac{1}{\sinh z}\).

Step 4: Differentiate the Inner Function

The derivative of \(g(z) = \sinh z\) with respect to \(z\) is \(g'(z) = \cosh z\). This is because the derivative of \(\sinh z\) is \(\cosh z\), by the definition of hyperbolic functions.

Step 5: Combine the Derivatives

Apply the chain rule: \(\frac{dy}{dz} = \frac{1}{\sinh z} \cdot \cosh z\). This simplifies to: \(\frac{dy}{dz} = \frac{\cosh z}{\sinh z}\).

Step 6: Simplify Further if Necessary

The expression \(\frac{\cosh z}{\sinh z}\) can be written as \(\coth z\), which is the hyperbolic cotangent function. Thus, the simplified form of the derivative is \(\frac{dy}{dz} = \coth z\).

Key Concepts

Natural Logarithm

The natural logarithm is often represented by the symbol \( \ln \). It is the logarithm to the base \( e \), where \( e \) is an irrational constant approximately equal to 2.71828. The natural logarithm is a key function in calculus and appears frequently in various mathematical contexts.

When you differentiate a natural logarithm, \( \ln(u) \), with respect to \( u \), the result is \( \frac{1}{u} \). This is because the natural logarithm is the inverse function of the exponential function. The differentiation rule stems from the fact that

• The derivative captures the rate of change.
• The slope of the tangent line to an exponential function at any point is proportional to the function's value at that point.
• This relationship leads to the derivative of \( \ln(u) \) being \( \frac{1}{u} \).

Understanding this derivative is crucial as it simplifies complex expressions and aids in solving differential equations.

Hyperbolic Functions

Hyperbolic functions, like trigonometric functions, arise naturally in certain mathematical situations. The hyperbolic sine function, \( \sinh(z) \), is defined as \( \sinh(z) = \frac{e^z - e^{-z}}{2} \). These functions are used in areas concerning hyperbolic geometry, calculus, and complex analysis.

The most interesting property of hyperbolic functions is their resemblance to trigonometric functions, but with some key differences. For instance, while \( \sinh(z) \) measures the 'height' in hyperbolic space, \( \cosh(z) \) measures the 'width', and their derivative relationships are quite simple:

• The derivative of \( \sinh(z) \) is \( \cosh(z) \).
• The derivative of \( \cosh(z) \) is \( \sinh(z) \).

Hyperbolic functions are key to solving certain types of differential equations and understanding exponential growth models in mathematical contexts.

Chain Rule

The chain rule is a fundamental tool in calculus used for differentiating composite functions. If you have a function that is the composition of two different functions, like \( y = f(g(z)) \), the chain rule provides the derivative of \( y \) with respect to \( z \).

The rule can be expressed as:\[\frac{dy}{dz} = f'(g(z)) \cdot g'(z)\]In simpler terms:

• Calculate the derivative of the outer function \( f(u) \) with respect to its input \( u \).
• Next, determine the derivative of the inner function \( g(z) \) with respect to \( z \).
• Multiply these two derivative results to obtain the derivative of the composite function.

The chain rule is particularly important when dealing with nested functions and allows you to differentiate functions in a step-by-step manner. This makes tackling more complex scenarios in physics, engineering, and economics more manageable.

Differentiation

Differentiation is the process of finding the derivative of a function. The derivative represents the rate of change of a function's value with respect to changes in its input. This is a central concept in calculus, which allows us to understand how functions behave locally.

Key aspects:

• It measures how a function's output changes as its input changes.
• Provides a means to find tangents, velocities, and slopes.
• Involves various rules such as the product rule, quotient rule, and chain rule to find derivatives of different types of functions.

Differentiation also plays a critical role in optimization, as it helps identify maximum and minimum values of functions. Whether it's finding the slope of a curve at a point, or optimizing costs in a business model, differentiation offers insights into a function's behavior and changes.