Question

Find the derivatives of the given functions and graph along with the function to ensure your answer is correct.[T] \(\frac{1}{\cosh (x)}\)

Short answer

The derivative of \( \frac{1}{\cosh(x)} \) is \(-\sech(x) \tanh(x)\).

Step-by-step solution

Understand the Function

The function given is \( f(x) = \frac{1}{\cosh(x)} \). It is the reciprocal of the hyperbolic cosine function.

Rewrite the Function

Rewrite \( f(x) \) using hyperbolic identities: \( f(x) = \sech(x) \), where \( \sech(x) = \frac{1}{\cosh(x)} \).

Use the Chain Rule

To differentiate \( f(x) = \sech(x) \), use the chain rule: \( \frac{d}{dx}[\sech(x)] = -\sech(x) \tanh(x) \).

Apply Derivative Formula

Using the derivative formula from Step 3, the derivative of \( f(x) = \sech(x) \) is given by: \( f'(x) = -\sech(x) \tanh(x) \).

Graph the Function and Its Derivative

Graph both \( f(x) = \sech(x) \) and its derivative \( f'(x) = -\sech(x) \tanh(x) \) to verify the behavior. Observe that the derivative is negative where the function decreases.

Key concepts

Hyperbolic Functions

Hyperbolic functions are analogs of trigonometric functions that are based on hyperbolas instead of circles. They include functions like hyperbolic sine (\( \sinh(x) \)), hyperbolic cosine (\( \cosh(x) \)), and hyperbolic tangent (\( \tanh(x) \)). These play a similar role in mathematics as the trigonometric functions, but are noted for their natural occurrence in many areas, such as calculus and complex analysis.

Unlike trigonometric functions that deal with geometric circles, hyperbolic functions relate to hyperbolic geometry. For instance, the hyperbolic cosine function, denoted \( \cosh(x) \), is defined as:\[\cosh(x) = \frac{e^x + e^{-x}}{2}.\]

Its reciprocal, which we focus on in this exercise, is the hyperbolic secant function, denoted as \( \sech(x) \), defined by:\[\sech(x) = \frac{1}{\cosh(x)}.\]

These functions are important because they help model real-world phenomena, such as the shape of a hanging cable or the distribution of a certain type of charge.

Chain Rule

The chain rule is a fundamental rule in calculus used to differentiate composite functions. It states that the derivative of a composite function \( f(g(x)) \) is given by the derivative of \( f \), evaluated at \( g(x) \), multiplied by the derivative of \( g \) itself. In formula terms, this is:\[\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x).\]

In the exercise, we have the function \( f(x) = \sech(x) \). To differentiate \( \sech(x) \) using the chain rule, we must consider \( \sech(x) \) as a composite function involving both the hyperbolic cosine and the hyperbolic tangent. The differentiation is applied as follows:\[\frac{d}{dx}[\sech(x)] = -\sech(x) \tanh(x).\]

Here, \( \tanh(x) = \frac{\sinh(x)}{\cosh(x)} \) is another hyperbolic function, and the chain rule tells us to multiply the derivative of \( \sech(x) \) by \( \tanh(x) \) to complete the process.

Derivative Graphing

Derivative graphing is a visual way to understand how a function behaves as it changes. When we graph a function and its derivative, we can observe patterns in the function's increase or decrease, and see the points where the function's slope changes.

For the function \( f(x) = \sech(x) \) and its derivative \( f'(x) = -\sech(x) \tanh(x) \), graphing helps us verify that the computations are correct. Unlike simple linear functions, hyperbolic functions can have interesting curve shapes, and their derivatives can provide key insights into their behavior.

When you graph \( \sech(x) \):

• The function decreases as \( x \) becomes large in either the positive or negative direction.
• The function has a maximum value of 1 when \( x = 0 \).

Graphing the derivative \( -\sech(x) \tanh(x) \):

• It is negative where the function \( \sech(x) \) decreases, confirming the behavior indicated by the derivative formula.

This visualization confirms our algebraic findings.

Reciprocal Functions

Reciprocal functions involve taking the reciprocal of a given function, meaning you invert its value. If \( f(x) \) is a non-zero function, then its reciprocal is \( \frac{1}{f(x)} \). The behavior of reciprocal functions often provides insights on the function's overall behavior and domain.

In the context of the exercise, the function \( f(x) = \frac{1}{\cosh(x)} \) is a classic example of a reciprocal function, reshaped to \( \sech(x) \). Reciprocal functions often have:

• Vertical asymptotes where the original function approaches zero, since division by zero is undefined.
• Symmetrical or inverse behavior since the reciprocal of positive numbers are positive, and the reciprocal of negative numbers are negative, but with magnitudes that are flipped.

The beauty of analyzing reciprocal functions lies in their ability to transform complex behaviors into something more interpretable, assisting both in theoretical mathematics and applied scenarios.