Problem 73 - Calculus Single Variable

Question

The hyperbolic tangent (tanh) and hyperbolic secant (sech) are defined by $$ \tanh (x)=\frac{\sinh (x)}{\cosh (x)}=\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}} $$ and $$ \operatorname{sech}(x)=\frac{1}{\cosh (x)}=\frac{2}{e^{x}+e^{-x}}. $$ Express \(\tanh ^{\prime}(x)\) and \(\operatorname{sech}^{\prime}(x)\) in terms of \(\tanh (x)\) and \(\operatorname{sech}(x)\)

Step-by-Step Solution

Answer

\( \tanh'(x) = \operatorname{sech}^2(x) \) and \( \operatorname{sech}'(x) = -\tanh(x) \cdot \operatorname{sech}(x) \).

Step 1: Differentiate Tanh Function

To find \( \tanh'(x) \), first recall that \( \tanh(x) = \frac{\sinh(x)}{\cosh(x)} \). Use the quotient rule for derivatives, which is given by: \[ \left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}, \] where \( u = \sinh(x) \) and \( v = \cosh(x) \). The derivatives are \( u' = \cosh(x) \) and \( v' = \sinh(x) \). Plug these into the quotient rule: \[ \tanh'(x) = \frac{\cosh(x)\cdot\cosh(x) - \sinh(x)\cdot\sinh(x)}{\cosh^2(x)}. \] This simplifies to: \[ \tanh'(x) = \frac{\cosh^2(x) - \sinh^2(x)}{\cosh^2(x)}. \]

Step 2: Simplify Tanh Derivative

Recall the identity \( \cosh^2(x) - \sinh^2(x) = 1 \), so the derivative becomes: \[ \tanh'(x) = \frac{1}{\cosh^2(x)}. \] But \( \cosh^2(x) = \frac{1}{\operatorname{sech}^2(x)} \), hence: \[ \tanh'(x) = \operatorname{sech}^2(x). \] This expresses \( \tanh'(x) \) in terms of \( \operatorname{sech}(x) \).

Step 3: Differentiate Sech Function

To find \( \operatorname{sech}'(x) \), consider \( \operatorname{sech}(x) = \frac{2}{e^x + e^{-x}} = \frac{1}{\cosh(x)} \). Using the quotient rule with \( u = 1 \) and \( v = \cosh(x) \), we have: \[ \operatorname{sech}'(x) = \frac{0 \cdot \cosh(x) - 1 \cdot \sinh(x)}{\cosh^2(x)}, \] which simplifies to: \[ \operatorname{sech}'(x) = -\frac{\sinh(x)}{\cosh^2(x)}. \]

Step 4: Simplify Sech Derivative

Recall that \( \tanh(x) = \frac{\sinh(x)}{\cosh(x)} \), so: \[ \sinh(x) = \cosh(x) \cdot \tanh(x). \] Substitute this into the derivative: \[ \operatorname{sech}'(x) = -\frac{\cosh(x) \cdot \tanh(x)}{\cosh^2(x)}. \] Simplify further: \[ \operatorname{sech}'(x) = -\tanh(x) \cdot \operatorname{sech}(x). \] This expresses \( \operatorname{sech}'(x) \) in terms of \( \tanh(x) \) and \( \operatorname{sech}(x) \).

Key Concepts

Hyperbolic Tangent

Hyperbolic tangent, often denoted as \( \tanh(x) \), is an important concept in mathematics, particularly in hyperbolic functions. It is defined by the equation \( \tanh(x) = \frac{\sinh(x)}{\cosh(x)} \). Here, \( \sinh(x) \) is the hyperbolic sine of \( x \), and \( \cosh(x) \) is the hyperbolic cosine of \( x \). These are essential functions that originate from exponential functions, giving us: \[ \tanh(x) = \frac{e^x - e^{-x}}{e^x + e^{-x}} \]Hyperbolic tangent is similar to the tangent function, but it applies to the hyperbolic functions, describing certain aspects of hyperbolas.

• It shows how quickly the hyperbolic angle's length grows compared to its width.
• It is typically used in calculus to study the behavior of curves and surfaces in geometry.
• The function is odd, meaning \( \tanh(-x) = -\tanh(x) \).

Understanding and differentiating \( \tanh(x) \) helps reveal the underlying behavior of other functions, such as the hyperbolic secant.

Hyperbolic Secant

The hyperbolic secant, written as \( \operatorname{sech}(x) \), is another key function when dealing with hyperbolic trigonometric functions. It is the reciprocal of the hyperbolic cosine:\[ \operatorname{sech}(x) = \frac{1}{\cosh(x)} = \frac{2}{e^x + e^{-x}} \]This function provides a way to understand how light intensity or wave amplitude diminishes over a certain distance in physical phenomena, often described by hyperbolic functions.

• Using \( \operatorname{sech}(x) \) can simplify complex expressions in calculus.
• It's valuable for solving differential equations in physics and engineering.
• Similar to the cosine, it reaches a maximum at \( x = 0 \) and decreases symmetrically as \( x \) increases or decreases.

Differentiating \( \operatorname{sech}(x) \) uses the quotient rule to express how \( \operatorname{sech}(x) \) changes. This derivative can show how the function diminishes or grows based on the value of \( x \).

Quotient Rule

The quotient rule is a key tool in calculus for finding the derivative of a quotient of two functions. It's expressed as:\[ \left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2} \]where \( u \) and \( v \) are functions of \( x \). The rule is especially useful for functions like \( \tanh(x) \) and \( \operatorname{sech}(x) \), which are expressed as quotients.

• The rule provides a systematic approach to find derivatives without simplifying the quotient first.
• It emphasizes multiplying the derivative of the numerator by the denominator and vice versa.
• This rule is handy for working with complex fractions and ratios that arise in advanced calculus.

In practical terms, the quotient rule helps break down complicated functions by focusing on change within the numerator and denominator separately, leading to insights into how the entire function behaves.

Derivatives of Hyperbolic Functions

Differentiating hyperbolic functions is fundamental in understanding their properties and applications. Just like their circular equivalents, hyperbolic functions have specific derivative rules.

• The derivative of \( \tanh(x) \), after applying the quotient rule and using certain trigonometric identities, simplifies to \( \operatorname{sech}^2(x) \).
• For \( \operatorname{sech}(x) \), its derivative is \(-\tanh(x) \cdot \operatorname{sech}(x) \).
• These derivatives provide algebraic expressions that describe how these functions change over different intervals.

These derivatives are crucial in more advanced mathematics, such as solving hyperbolic differential equations, and in real-world applications in physics and engineering. By utilizing these derivatives, students can better understand how hyperbolic functions behave similarly and differently from trigonometric functions.