Question

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

Short answer

The derivative of \( f(x) \) is \( \frac{2\text{sech}^2(x)}{(1-\tanh(x))^2} \), indicating the function is always increasing.

Step-by-step solution

Understand the Function

We are given the function \( f(x) = \frac{1 + \tanh(x)}{1 - \tanh(x)} \). Our goal is to find its derivative, \( f'(x) \), using rules of differentiation.

Apply the Quotient Rule

The quotient rule states that if we have a function \( \frac{u}{v} \), then its derivative is \( \frac{u'v - uv'}{v^2} \). Here, \( u = 1 + \tanh(x) \) and \( v = 1 - \tanh(x) \).

Differentiate the Numerator and Denominator

Calculate \( u' \) and \( v' \):- \( u' = \frac{d}{dx}(1 + \tanh(x)) = \text{sech}^2(x) \)- \( v' = \frac{d}{dx}(1 - \tanh(x)) = -\text{sech}^2(x) \)

Substitute into the Quotient Rule Formula

Substitute \( u, u', v, v' \) into the quotient rule:\[\frac{d}{dx}\left(\frac{1 + \tanh(x)}{1 - \tanh(x)}\right) = \frac{\text{sech}^2(x) \cdot (1 - \tanh(x)) - (1 + \tanh(x)) \cdot (-\text{sech}^2(x))}{(1 - \tanh(x))^2}\]

Simplify the Derivative Expression

Simplify the expression:\[= \frac{\text{sech}^2(x) + 2\tanh(x)\text{sech}^2(x) + \tanh^2(x)\text{sech}^2(x)}{(1 - \tanh(x))^2}\]

Simplify Further and Result Interpretation

Factor out \( \text{sech}^2(x) \): \[= \frac{\text{sech}^2(x)(1 + 2\tanh(x) + \tanh^2(x))}{(1 - \tanh(x))^2}\]Express \( (1 + \tanh(x))^2 \) using identity \( 1 - \tanh^2(x) = \text{sech}^2(x) \), leading to:\[= \frac{1}{(1-\tanh(x))^2}\]

Graph and Verification

Graph both \( f(x) = \frac{1 + \tanh(x)}{1 - \tanh(x)} \) and its derivative.Observing that the derivative is never zero and approaches infinity, confirms that the function always increases.

Key concepts

Quotient Rule

The quotient rule in calculus is a method used to differentiate functions that are divided by one another. Imagine you're working with a fraction where the numerator and the denominator are both functions. If you want to find the derivative of this fraction, the quotient rule is here to help. It states: if you have a function expressed as \( \frac{u}{v} \), the derivative, \( \frac{d}{dx}(\frac{u}{v}) \), is calculated as \( \frac{u'v - uv'}{v^2} \).

In this rule, \(u\) represents the numerator, and \(\v\) represents the denominator. The terms \(u'\) and \(v'\) are the derivatives of \(u\) and \(v\) respectively. Applying the quotient rule involves four main pieces: find the derivative of the numerator, multiply it by the denominator, then do the opposite for the numerator, and subtract those results. Finally, divide by the square of the denominator.

• Calculate the derivative of the numerator \(u'\)
• Calculate the derivative of the denominator \(v'\)
• Apply the formula: \(u'v - uv'\)
• Divide by \(v^2\)

Hyperbolic Functions

Hyperbolic functions include hyperbolic sine \( \sinh(x) \) and hyperbolic cosine \( \cosh(x) \), among others. They are analogous to trigonometric functions but relate more closely to exponential functions. In our exercise, we encounter \( \tanh(x) \), the hyperbolic tangent.

The hyperbolic tangent function is expressed as \( \tanh(x) = \frac{\sinh(x)}{\cosh(x)} \). Its derivative is \( \text{sech}^2(x) \). This is crucial when differentiating expressions involving \( \tanh(x) \).

Hyperbolic functions often arise in mathematical physics and engineering. Their applications can range from representing shapes such as catenary curves (seen in hanging cables) to solving hyperbolic equations which describe wave phenomena.

• Understanding \( \tanh(x) \) and its derivatives can provide insights into behavior across varied applications.

Graphical Analysis

Graphical analysis involves looking at the graph of a function to gain insight into its behavior. After calculating derivatives, plotting the original function and its derivative can be very enlightening.

For \( f(x) = \frac{1 + \tanh(x)}{1 - \tanh(x)} \), once we find the derivative, we consider its root and range. Observing the derivative tells us about the slope and rate of change of the function. If the derivative is always positive, the function is always increasing.

With graphical analysis, try to notice:

• Where the function doesn't change direction (no zero derivatives)
• How steep the curve is, showing rapid changes
• Any asymptotic behavior, indicating points the function approaches but never reaches

Graphical analysis helps confirm analytical derivative findings by showing visually that our calculations make sense in a broader context.