Problem 380 - Calculus Volume 1

Question

Use the quotient rule to verify that \(\tanh (x)^{\prime}=\operatorname{sech}^{2}(x)\)

Step-by-Step Solution

Answer

The derivative of \( \tanh(x) \) is \( \operatorname{sech}^2(x) \).

Step 1: Recall the Definition of Hyperbolic Tangent

The function we want to differentiate is the hyperbolic tangent, which is defined as \( \tanh(x) = \frac{\sinh(x)}{\cosh(x)} \). This is a quotient of two functions, the hyperbolic sine \( \sinh(x) \) and the hyperbolic cosine \( \cosh(x) \).

Step 2: Apply the Quotient Rule

The quotient rule states that for two differentiable functions \( u(x) \) and \( v(x) \), the derivative of the quotient \( \frac{u}{v} \) is \( \left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2} \). Apply this to \( \tanh(x) \):\((u, v) = (\sinh(x), \cosh(x))\).The derivatives are \( u' = \cosh(x) \) and \( v' = \sinh(x) \).

Step 3: Differentiate Using the Quotient Rule

Substitute \( u, u', v, \) and \( v' \) into the quotient rule formula:\[ \left( \frac{\sinh(x)}{\cosh(x)} \right)' = \frac{\cosh(x) \times \cosh(x) - \sinh(x) \times \sinh(x)}{\cosh^2(x)} \]

Step 4: Simplify the Expression

Simplify the numerator using the identity \( \cosh^2(x) - \sinh^2(x) = 1 \) to get:\[ \frac{1}{\cosh^2(x)} \]

Step 5: Recognize the Result as Hyperbolic Secant Squared

The expression \( \frac{1}{\cosh^2(x)} \) is the definition of the square of the hyperbolic secant function: \( \operatorname{sech}^2(x) \). Thus, we have shown that:\[ \tanh(x)' = \operatorname{sech}^2(x) \]

Key Concepts

Hyperbolic Functions

Hyperbolic functions 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 are analogous to ordinary trigonometric functions. They are useful in describing

hyperbolas

and solving certain differential equations

available in calculus and physics.Key properties of hyperbolic functions include:

• Even and odd functions: \(\cosh(x)\) is even, while \(\sinh(x)\) and \(\tanh(x)\) are odd.
• The identity \(\cosh^2(x) - \sinh^2(x) = 1\), much like the Pythagorean identity for trigonometric functions.

Understanding these ensures you can apply them effectively in various mathematical and scientific scenarios.

Derivatives

Derivatives measure how a function changes as its input changes. They are fundamental in calculus to determine the rate of change or slope of a curve. For example, the derivative \(\tanh(x)'\) is found using the quotient rule. In calculus, understanding how to compute derivatives of different types of functions efficiently, like hyperbolic ones, allows you to

• analyze graphs
• predict future behavior of systems
• and solve optimization problems

effectively. For a quotient of two functions, \(\frac{u}{v}\), the derivative is derived as follows:- Take the derivative of the numerator \(u'\)- Multiply it by the denominator \(v\)- Subtract the product of the original numerator \(u\) and the derivative of the denominator \(v'\)- Divide the entire expression by the square of the original denominator \(v^2\).

Simplification

Simplification involves turning complex expressions into simpler forms that are easier to understand or evaluate. In the context of derivatives, simplification often reveals deeper connections between functions. When differentiating \(\tanh(x)\) using the quotient rule, you obtain:every element involved in the process: \(\cosh(x)^2 - \sinh(x)^2\) Using the identity \(\cosh^2(x) - \sinh^2(x) = 1\) to simplify the numerator,allows you to transform your derivative into
\(\frac{1}{\cosh^2(x)}\).
Such simplification leads directly to a recognizable form, \(\operatorname{sech}^2(x)\), illustrating the power of simplification in calculus.

Calculus

Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. It is divided into differential calculus and integral calculus. - Differential calculus concerns itself with the notion of the derivative, which is essentially about rates of change. In the exercise given, you differentiated the hyperbolic tangent function.
- Integral calculus, on the other hand, is concerned primarily with the concept of accumulation and area under curves. Calculus provides the foundational language and tools necessary to model and analyze dynamic systems in science and engineering. It helps predict movement, understand change, and find optimal solutions in various fields. Understanding calculus, including derivatives of less intuitive functions like hyperbolic ones, allows individuals to confidently navigate through complex problems and predict system behaviors efficiently.