Problem 392 - Calculus Volume 1

Question

For the following exercises, find the derivatives of the given functions and graph along with the function to ensure your answer is correct. $$\frac{1+\tanh (x)}{1-\tanh (x)}$$

Step-by-Step Solution

Answer

The derivative is \( f'(x) = \frac{2 \cdot \text{sech}^2(x)}{(1 - \tanh(x))^2} \).

Step 1: Identify the Function

The function to differentiate is given as \( f(x) = \frac{1 + \tanh(x)}{1 - \tanh(x)} \). We will need to apply the quotient rule to find the derivative.

Step 2: Recall the Derivative Formulas

Recall that the derivative of \( \tanh(x) \) is \( \text{sech}^2(x) \). The quotient rule for derivatives states that the derivative of a function \( \frac{u}{v} \) is \( \frac{u'v - uv'}{v^2} \), where \( u \) and \( v \) are functions of \( x \).

Step 3: Apply the Quotient Rule

Let \( u = 1 + \tanh(x) \) and \( v = 1 - \tanh(x) \). Then \( u' = \text{sech}^2(x) \) and \( v' = -\text{sech}^2(x) \). The derivative of \( f(x) \) is given by:\[ f'(x) = \frac{u'v - uv'}{v^2} = \frac{(\text{sech}^2(x))(1 - \tanh(x)) - (1 + \tanh(x))(-\text{sech}^2(x))}{(1 - \tanh(x))^2} \].

Step 4: Simplify the Derivative

Simplify the derivative expression:\[ f'(x) = \frac{\text{sech}^2(x)(1 - \tanh(x) + 1 + \tanh(x))}{(1 - \tanh(x))^2} = \frac{2 \cdot \text{sech}^2(x)}{(1 - \tanh(x))^2} \].Notice that \( \tanh^2(x) + \text{sech}^2(x) = 1 \), simplifying further if needed.

Step 5: Graph the Function and Derivative

Using graphing software or a graphing calculator, plot the original function \( f(x) \) and its derivative \( f'(x) = \frac{2 \cdot \text{sech}^2(x)}{(1 - \tanh(x))^2} \). Observe that the derivative correctly reflects the slope behavior of the original function.

Key Concepts

Quotient Rule

In calculus, when you're dealing with the derivative of a quotient of two functions, the quotient rule is your go-to tool. This rule is essential to differentiate functions that are written as one function divided by another.
To apply the quotient rule, you have to identify two parts, say, \( u \) and \( v \), which are functions of \( x \). The rule is stated as:

• If \( f(x) = \frac{u}{v} \), then \( f'(x) = \frac{u'v - uv'}{v^2} \).

Here's how it breaks down:

• First, compute \( u' \), the derivative of the numerator.
• Then, find \( v' \), the derivative of the denominator.
• Apply them into the formula \( \frac{u'v - uv'}{v^2} \).

The result is a formula that gives you the derivative of the original function expressed as a ratio of derivatives and original functions. This ensures you can handle complex-looking functions with ease.

Hyperbolic Functions

Hyperbolic functions, like their trigonometric counterparts, are indispensable in calculus for describing certain types of curves. Key hyperbolic functions include \( \sinh(x) \), \( \cosh(x) \), and \( \tanh(x) \), along with their derivatives. An essential aspect of these functions is that they have properties similar to familiar trigonometric functions.
The hyperbolic tangent, \( \tanh(x) \), is a key function here, often appearing in calculus problems due to its nature of describing hyperbolic angles and their transformations. Its derivative, the \( \text{sech}^2(x) \), behaves similarly to the square of the secant function in trigonometry, being also closely related with hyperbolic cosine, \( \cosh(x) \), by the relation \( \cosh^2(x) - \sinh^2(x) = 1 \).

• These properties make hyperbolic functions crucial in simplifying complex derivatives.
• Provide a symmetrical understanding akin to trigonometric identities.

Using these functions and their derivatives propels you further into advanced calculus concepts, allowing for a deeper comprehension of continuous and smooth transformations.

Graphing Derivatives

Graphing derivatives is a powerful way to visually understand the behavior of functions and their rates of change. By comparing graphs of functions and their derivatives, one can see how the slope of the tangent line to the function varies over its domain.
When graphing the derivative of a function like \( f(x) = \frac{1 + \tanh(x)}{1 - \tanh(x)} \):

• First, determine the derivative using calculus—the quotient rule, in this case.
• Next, plot the original function's graph over the desired interval using computational tools or graph paper.
• Then overlay the derivative graph, \( f'(x) = \frac{2 \cdot \text{sech}^2(x)}{(1 - \tanh(x))^2} \), onto the same axes.

This graphical representation allows us to confirm our mathematical results by observing where the function is increasing or decreasing. Graphs reveal characteristics such as maximums, minimums, and points of inflection—information that's sometimes challenging to ascertain from equations alone.

Sech Function

The hyperbolic secant function, or \( \text{sech}(x) \), plays a crucial role when dealing with hyperbolic functions. It is the reciprocal of the hyperbolic cosine: \( \text{sech}(x) = \frac{1}{\cosh(x)} \).
The significance of \( \text{sech}(x) \) in calculus emerges prominently when calculating derivatives, especially with the derivative of hyperbolic tangent being \( \text{sech}^2(x) \). Here are a few important points:

• \( \text{sech}(x) \) simplifies many expressions involving \( \tanh(x) \).
• Understanding its behavior is key in problems involving hyperbolic functions.
• The function decreases as \( x \) moves away from zero, as \( \cosh(x) \) becomes very large.

Emphasizing the role of \( \text{sech}(x) \) in derivatives helps students appreciate its utility in addressing and simplifying calculus problems involving hyperbolic expressions.