Problem 18 - Essential Calculus

Question

Find the derivative of the function. \(g(x)=\ln (\cosh x)\)

Step-by-Step Solution

Answer

The derivative of the function \(g(x) = \ln (\cosh x)\) is \(g'(x) = \tanh x\).

Step 1: Identify the inner function

First, recognize that \(g(x) = \ln (\cosh x)\) is comprised of an outer function, the natural logarithm \(\ln (x)\), and an inner function, the hyperbolic cosine \(\cosh x\). Thus, the function is composed of two other functions.

Step 2: Apply the chain rule

To differentiate this composite function, the chain rule will be implemented: \((f(g(x)))' = f'(g(x)) \cdot g'(x)\). For this case, \(f(x) = \ln x\) and \(g(x) = \cosh x\). From the chain rule, the derivative of the function \(g(x)\) is: \(g'(x) = \frac{1}{\cosh x} \cdot (\cosh x)'\).

Step 3: Compute the derivative of \(\cosh x\)

The derivative of the hyperbolic cosine \(\cosh x\) is \(\sinh x\). Thus, \(g'(x) = \frac{1}{\cosh x} \cdot \sinh x\).

Step 4: Simplify the expression

The quotient \(\frac{\sinh x}{\cosh x}\) equals to \(\tanh x\), thus the derivative of the function \(g(x)\) is fully simplified as: \(g'(x) = \tanh x\).

Key Concepts

Chain Rule

The chain rule is a fundamental concept in calculus used to differentiate composite functions. Imagine stacking several processes, like peeling an apple and then slicing it. The chain rule helps us find how fast the outer process changes by considering the speed of each inner step. This is crucial when a function is made from two or more functions nested within each other.

To apply the chain rule, you follow this simple idea: first, differentiate the outer function while keeping the inner function intact, then multiply it by the derivative of the inner function. Mathematically, if you have a function \(f(g(x))\), the chain rule formula is:

• \((f(g(x)))' = f'(g(x)) \cdot g'(x)\)

In our exercise, the function has an outer part, \(\ln(x)\), and an inner part, \(\cosh(x)\). When you differentiate \(\ln(\cosh(x))\), you're applying the chain rule. First, find the derivative of \(\ln(x)\) with respect to \(x\), which is \(\frac{1}{x}\), while keeping \(\cosh(x)\) as is. Then multiply this by the derivative of \(\cosh(x)\).

The chain rule enables us to "chain" together the differentiation process for complex functions, making it a powerful tool.

Derivative

A derivative, in calculus, tells us how a function changes as its input changes – essentially, it's the function's rate of change or its slope at any given point. Imagine driving a car: the speedometer shows how fast you're going at each instant. That's akin to the derivative of your distance over time.

In the exercise, we're tasked with finding the derivative of \(g(x) = \ln(\cosh(x))\). Each component of this function has its own behaviour and rate of change. To tackle such functions, understanding each part's instantaneous rate of change is essential. The goal is to determine \(g'(x)\), which will reveal how \(g(x)\) changes as \(x\) changes.

• Firstly, differentiate \(f(x) = \ln(x)\). Its derivative is \(\frac{1}{x}\).
• Next, differentiate the inner function, \(g(x) = \cosh(x)\), yielding \(g'(x) = \sinh(x)\).
• Apply these derivatives using the chain rule to find the rate at which \(g(x)\) changes.

Finding derivatives is essential for graphing functions, solving real-world problems, and understanding dynamic systems.

Hyperbolic Functions

Hyperbolic functions are analogues of trigonometric functions but for hyperbolas, much like how sine and cosine relate to circles. They're incredibly useful in various fields of science and engineering, where ideal conditions are non-circular. Two of these functions are the hyperbolic cosine, \(\cosh(x)\), and hyperbolic sine, \(\sinh(x)\).

Hyperbolic cosine is defined as:

• \(\cosh(x) = \frac{e^x + e^{-x}}{2}\)

These functions describe a range of natural phenomena, such as the shape of hanging cables or the growth of certain populations.

In our solution, we differentiate \(\cosh(x)\), yielding \(\sinh(x)\), which emphasises the beautifully interconnected nature of hyperbolic functions. Just like \(\cos(x)\) and \(\sin(x)\) in trigonometry, \(\cosh(x)\) and \(\sinh(x)\) have a close mathematical relationship.

• The derivative of \(\cosh(x)\) is \(\sinh(x)\), showing a similar interplay to trigonometric functions where \(\frac{d}{dx} \cos(x) = -\sin(x)\).

Understanding these functions and their derivatives is crucial for solving complex calculus problems, modelling real-world phenomena, and conducting precise engineering analyses.