Problem 391 - 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. $$\tanh \left(\sqrt{x^{2}+1}\right)$$

Step-by-Step Solution

Answer

Derive: \( f'(x) = \text{sech}^2(\sqrt{x^2 + 1}) \cdot \frac{x}{\sqrt{x^2 + 1}} \).

Step 1: Identify the function's components

The function given is \( f(x) = \tanh (\sqrt{x^2 + 1}) \). Here, \( \tanh(u) \) is the hyperbolic tangent function, where \( u = \sqrt{x^2 + 1} \). We need to find the derivative \( f'(x) \).

Step 2: Use the Chain Rule

To find the derivative, we apply the chain rule. The chain rule states that if \( f(x) = g(h(x)) \), then the derivative \( f'(x) = g'(h(x)) \cdot h'(x) \). Here, \( g(u) = \tanh(u) \) and \( h(x) = \sqrt{x^2 + 1} \).

Step 3: Differentiate \( \tanh(u) \)

The derivative of \( \tanh(u) \) with respect to \( u \) is \( \text{sech}^2(u) \), where \( \text{sech}(u) \) is the hyperbolic secant function, defined as \( 1/\cosh(u) \). So, \( g'(u) = \text{sech}^2(u) \).

Step 4: Differentiate \( \sqrt{x^2 + 1} \)

The function \( u = \sqrt{x^2 + 1} \) can be rewritten as \( (x^2 + 1)^{1/2} \). Its derivative with respect to \( x \) is \( u' = \frac{x}{\sqrt{x^2 + 1}} \) using the power rule.

Step 5: Apply the Chain Rule

Combine the results from Steps 3 and 4 with the chain rule: \[ f'(x) = \text{sech}^2(\sqrt{x^2 + 1}) \cdot \frac{x}{\sqrt{x^2 + 1}} \] This is the derivative of the function.

Step 6: Graph the Function and Derivative

Graph the original function \( \tanh(\sqrt{x^2 + 1}) \) and its derivative \( f'(x) = \text{sech}^2(\sqrt{x^2 + 1}) \cdot \frac{x}{\sqrt{x^2 + 1}} \) to visually verify that the derivative correctly corresponds to the rate of change of the function. The derivative should show the rate at which \( f(x) \) changes for different values of \( x \).

Key Concepts

Derivative

In the realm of calculus, derivatives are crucial because they tell us how a function changes. They represent the rate at which one quantity changes with respect to another. When we are dealing with a function, the derivative at a given point shows the slope of the tangent to the curve at that point. This can help us understand how the function behaves in small intervals around the point.
To find the derivative of a complex function, especially one involving hyperbolic functions, we often need to employ techniques like the chain rule. By differentiating, we can gain insights into the function's critical points, inflection points, and more, making the function's graph simpler to analyze.

Hyperbolic functions

Hyperbolic functions are similar to trigonometric functions but involve hyperbolas instead of circles. One of the most common hyperbolic functions is the hyperbolic tangent, denoted as \( \tanh(x) \). It's defined as \( \tanh(x) = \frac{\sinh(x)}{\cosh(x)} \), where \( \sinh(x) \) and \( \cosh(x) \) are the hyperbolic sine and cosine functions, respectively.
Hyperbolic tangent functions have unique properties. They range between -1 and 1, showing characteristics similar to the arctangent function regarding its horizontal asymptotes. These functions are notably used in various applications, including the modeling of growth processes, signal processing, and in solving hyperbolic differential equations.
The derivative of the hyperbolic tangent function is \( \text{sech}^2(x) \), where \( \text{sech}(x) \) is the hyperbolic secant. Hence, knowing the derivative provides us with more information on how rapidly the \( \tanh \) function changes.

Chain rule

The chain rule is pivotal when it comes to differentiating composite functions. It offers a systematic way to break down the process of finding the derivatives of functions nested within each other. Essentially, if a function \( f(x) \) is composed of a function \( g(x) \) within another function \( h(x) \), like \( f(x) = g(h(x)) \), we can find its derivative using the chain rule.
The chain rule states: \( f'(x) = g'(h(x)) \cdot h'(x) \). This means you first find the derivative of the outer function evaluated at the inner function, and then multiply it by the derivative of the inner function.
For the function \( \tanh(\sqrt{x^2 + 1}) \), the tanh function is the outer function, and \( \sqrt{x^2 + 1} \) is the inner one. By applying the chain rule, we can efficiently determine the derivative, which guides us in understanding the nuanced behaviors of the original function.

Graphing functions

Graphing functions and their derivatives serves as a vital tool in calculus. It visually represents how a function behaves over different domains. When you graph a function like \( \tanh(\sqrt{x^2 + 1}) \) alongside its derivative, the graph will provide a tangible comparison of the function’s slope at various points.
The derivative graph shows where the function increases, decreases, or remains constant. For instance, where the derivative crosses the x-axis at zero, the original function hits a local extremum or inflection point. Employing graphing techniques allows students to substantiate their calculus solutions and deepen their comprehension.
Graphing also simplifies spotting symmetry, asymptotic behavior, and periodicity within the function. It's a practical approach that turns abstract calculus problems into understandable visual stories. This enhances learning and offers a clear, interpretable representation of how functions and their derivatives intertwine.