Problem 4 - Calculus

Question

By the quotient rule, verify \((\tanh x)^{\prime}=\operatorname{sech}^{2} x\).

Step-by-Step Solution

Answer

The derivative \( (\tanh x)' = \operatorname{sech}^2 x \) is verified using the quotient rule and known identities.

Step 1: Recall the definition

The hyperbolic tangent is defined as follows: \[ \tanh x = \frac{\sinh x}{\cosh x} \]where \( \sinh x \) and \( \cosh x \) are the hyperbolic sine and cosine functions respectively.

Step 2: Apply the Quotient Rule

The derivative of a quotient \( \frac{u}{v} \) is given by\[ \left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2} \]For \( \tanh x = \frac{\sinh x}{\cosh x} \), we identify \( u = \sinh x \) and \( v = \cosh x \). Thus, we need to find \[ \left( \tanh x \right)' = \frac{(\sinh x)' \cdot \cosh x - \sinh x \cdot (\cosh x)'}{(\cosh x)^2} \].

Step 3: Differentiate the numerator and denominator

The derivatives of the hyperbolic functions are:\[ (\sinh x)' = \cosh x \]\[ (\cosh x)' = \sinh x \]Substitute these derivatives back into the quotient rule formula:\[ \frac{(\cosh x) \cdot \cosh x - \sinh x \cdot \sinh x}{(\cosh x)^2} \].

Step 4: Simplify the expression

Simplify the expression: \[ (\cosh x)^2 - (\sinh x)^2 \].Using the identity \( (\cosh x)^2 - (\sinh x)^2 = 1 \), the expression simplifies to:\[ \frac{1}{(\cosh x)^2} \].

Step 5: Identify the result as \( \operatorname{sech}^2 x \)

Recognize that \( \operatorname{sech}^2 x = \frac{1}{\cosh^2 x} \).Thus, the derivative \[ (\tanh x)' = \operatorname{sech}^2 x \], confirming the given statement.

Key Concepts

Quotient Rule

The Quotient Rule is a fundamental concept in calculus used to find the derivative of the division of two functions. Given two differentiable functions, say \( u(x) \) and \( v(x) \), the Quotient Rule provides a way to compute the derivative of their quotient \( \frac{u}{v} \). The formula is expressed as:\[\left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2}\]Here's a simple way to remember it:

• Differentiation of the top function (numerator)
• Multiply by the original bottom function (denominator)
• Subtract the product of the original top function and the derivative of the bottom
• Divide the whole expression by the square of the bottom function

Keeping these steps in mind can simplify finding derivatives of quotients. Always check that both functions \( u(x) \) and \( v(x) \) are differentiable to apply this rule.

Hyperbolic Functions

Hyperbolic functions, much like trigonometric functions, emerge from hyperbolas rather than circles. These functions are essential in various fields such as calculus, engineering, and physics due to their unique properties.There are several hyperbolic functions, but the most common ones include:

• Hyperbolic Sine (\( \sinh x = \frac{e^x - e^{-x}}{2} \)
• Hyperbolic Cosine (\( \cosh x = \frac{e^x + e^{-x}}{2} \)
• Hyperbolic Tangent (\( \tanh x = \frac{\sinh x}{\cosh x} \)

These functions have profound implications across various problems, particularly in the field of calculus. One important identity with hyperbolic functions is:\[\cosh^2 x - \sinh^2 x = 1\]This identity is crucial because it helps simplify expressions and verify results, such as confirming the derivative of hyperbolic tangent. Approaching hyperbolic functions with the same mindset as trigonometric functions can help in grasping their complexity.

Derivative

The derivative is one of the cornerstone concepts in calculus. It represents how a function changes as its input changes, essentially describing the function's rate of change or slope at any given point.When finding the derivative of a function, we are following the processes of:

• Calculating the limit of the function's average rate of change as the change in input approaches zero
• Using different rules, such as the Power Rule, Product Rule, and Quotient Rule, to find this rate of change efficiently
• Applying fundamental identities and relationships to simplify results

Understanding derivatives provides powerful insights into the behavior and trends of functions. In the context of hyperbolic functions, discovering the derivative of \( \tanh x \), such as in the example previously discussed, involves using the Quotient Rule.By applying the Quotient Rule and knowing specific derivatives for \( \sinh x \) and \( \cosh x \), we can derive results such as \( (\tanh x)' = \operatorname{sech}^2 x \). This derivative simplifies analysis of changes in hyperbolic functions and often appears in practical engineering or physics applications.