Problem 390 - 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. $$\cosh ^{2}(x)-\sinh ^{2}(x)$$

Step-by-Step Solution

Answer

The derivative is 0, as \( \cosh^2(x) - \sinh^2(x) = 1 \).

Step 1: Recall Hyperbolic Identity

The identity for hyperbolic cosines and sines is: \( \cosh^2(x) - \sinh^2(x) = 1 \). This is similar to the Pythagorean identity for trigonometric functions, and it holds for all real numbers \( x \). Based on this identity, the expression simplifies to 1. It means \( \cosh^2(x) - \sinh^2(x) \) is actually a constant function.

Step 2: Differentiate the Constant Function

Since the expression simplifies to a constant, namely 1, its derivative is 0. The derivative of any constant function is 0, regardless of the input variable.

Step 3: Graph the Functions and Verify

Graph the function \( y = \cosh^2(x) - \sinh^2(x) \) and the constant function \( y = 1 \) to verify that they are identical. Both should appear as horizontal lines at the same level, confirming that the derivative is correct as any constant's slope is zero.

Key Concepts

Hyperbolic Functions

Hyperbolic functions can seem mysterious at first, but they are quite similar to the trigonometric functions you're likely more familiar with. Instead of circles, hyperbolic functions relate to hyperbolas—a different kind of curve. Two primary hyperbolic functions are the hyperbolic sine (\( \sinh(x) \)) and the hyperbolic cosine (\( \cosh(x) \)).
Just like their circular counterparts, these functions have useful properties and applications in calculus, particularly in engineering and physics.
The hyperbolic sine function is defined as:

• \( \sinh(x) = \frac{e^x - e^{-x}}{2} \)

The hyperbolic cosine function is defined as:

• \( \cosh(x) = \frac{e^x + e^{-x}}{2} \)

Much like sine and cosine, \( \sinh(x) \) and \( \cosh(x) \) have similar properties such as identities and derivatives, aiding in simplifying complex functions and solving calculus problems.

Hyperbolic Identities

In calculus, special identities simplify the process of differentiation and integration. For hyperbolic functions, one of the most essential identities is the following:

• \( \cosh^2(x) - \sinh^2(x) = 1 \)

This equation mirrors the structure of the famous Pythagorean identity \( \cos^2(x) + \sin^2(x) = 1 \).
To understand this identity, consider the parabolic nature of these functions. When evaluating \( \cosh(x) \) and \( \sinh(x) \) at any value of \( x \), their squared difference will always equal one. This is not just a random fact, but a result of their backbones being formed by the exponential function. Hyperbolic identities such as this are handy when simplifying expressions and determining derivatives, as seen with the function \( \cosh^2(x) - \sinh^2(x) \) which simply equals the constant 1.

Differentiating Constant Functions

Differentiation can sometimes look like a daunting task, but when you're dealing with a constant function, like \( \cosh^2(x) - \sinh^2(x) = 1 \), it's quite straightforward. The derivative of a constant, regardless if it is 1, 10, or any other number, is always 0. Why is that? Because a constant's value doesn't change as \( x \) changes.
In essence, when you graph such a function, it appears as a horizontal line on a graph. The slope of this line, key to understanding derivatives, is zero because there is no vertical change, no matter how much you "zoom out" or "zoom in." This knowledge is powerful, particularly when analyzing the behavior of functions in calculus.
To differentiate a constant function:

• Recognize the constant, here it's 1.
• Understand that its derivative is 0.
• Visualize the graph to verify correctness, as it will run parallel to the x-axis throughout infinity.

Understanding this basic principle helps create a firm foundation for tackling more complex calculus problems that involve combinations of constants and variable parts.