Problem 7 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find \(D_{x} y\). $$ y=\ln (\sinh x) $$

Step-by-Step Solution

Answer

The derivative is \(\coth(x)\).

Step 1: Identify the Rule

To find the derivative of a logarithmic function like \(y=\ln(f(x))\), we apply the chain rule and the derivative of natural logarithm: \(\frac{d}{dx}\ln(f(x)) = \frac{1}{f(x)} \cdot f'(x)\). In this case, \(f(x) = \sinh(x)\).

Step 2: Differentiate \(\sinh(x)\)

First, find the derivative of \(\sinh(x)\). The derivative of \(\sinh(x)\) is \(\cosh(x)\).

Step 3: Apply the Chain Rule

Using the chain rule for the derivative of \(\ln(f(x))\), substitute \(f(x) = \sinh(x)\) and \(f'(x) = \cosh(x)\). So, we get: \(D_{x} y = \frac{1}{\sinh(x)} \cdot \cosh(x)\).

Step 4: Simplify the Expression

Simplify the expression \(\frac{\cosh(x)}{\sinh(x)}\). By the definition of hyperbolic functions, this can be simplified to \(\coth(x)\). Thus, \(D_{x} y = \coth(x)\).

Key Concepts

Chain Rule

The chain rule is a fundamental technique used in calculus, especially when differentiating composite functions. It tells us how to differentiate a function of a function, or more formally, if you have a composition of two functions, say, \( g(f(x)) \). The derivative is found by multiplying the derivative of the outer function by the derivative of the inner function.

Imagine you have a situation like \( y = \ln(\sinh(x)) \). Here, the outer function is the natural logarithm \( \ln \), and the inner function is the hyperbolic sine function \( \sinh(x) \).

In mathematical terms, if \( y = u(v(x)) \), then the derivative \( \frac{dy}{dx} \) is \( u'(v(x)) \cdot v'(x) \). This multiplication arises because as one function changes, it affects the change in the other function as well.

Once you grasp this concept, applying it to other functions becomes much easier and intuitive.

Logarithmic Differentiation

Logarithmic differentiation is a powerful tool, especially handy when you are dealing with complex products, quotients, or powers of functions. The rule for differentiating \( \ln(f(x)) \) is simply \( \frac{1}{f(x)} \cdot f'(x) \). This approach uses the properties of logarithms to transform multiplication into addition and division into subtraction, making differentiation more manageable.

When applied to our example, \( y = \ln(\sinh(x)) \), you observe that it's about finding how the function \( \sinh(x) \) behaves as it passes through \( \ln \). By using logarithmic differentiation, we easily end up with \( \frac{1}{\sinh(x)} \cdot \cosh(x) \), which is its derivative.

This method is very effective as it simplifies differentiating even the most complicated expressions, making them much more digestible.

Hyperbolic Functions

Hyperbolic functions are analogs of trigonometric functions but for a hyperbola instead of a circle. The basic hyperbolic functions you encounter are \( \sinh(x) \) and \( \cosh(x) \). These functions frequently appear in calculus and differential equations.

The function \( \sinh(x) = \frac{e^x - e^{-x}}{2} \) is quite similar to the sine function, while \( \cosh(x) = \frac{e^x + e^{-x}}{2} \) parallels the cosine function.

• \( \sinh(x) \), the hyperbolic sine, grows exponentially as x increases.
• \( \cosh(x) \), the hyperbolic cosine, remains positive and represents a hyperbolic version of the cosine wave.

When you compute their derivatives:

• The derivative of \( \sinh(x) \) is \( \cosh(x) \).
• And, interestingly, the derivative of \( \cosh(x) \) is \( \sinh(x) \).

This mutual relationship makes calculations involving these functions both intriguing and straightforward once the patterns are understood.