Question

Find the derivatives of the given functions and graph along with the function to ensure your answer is correct.[T] ${\cosh }^2(x)-{\sinh }^2(x)$

Short answer

The derivative is 0; both graphs confirm this as they are constant lines.

Step-by-step solution

Simplify the Expression

The function is given as ${\cosh }^2(x)-{\sinh }^2(x)$. We know from hyperbolic identities that ${\cosh }^2(x)-{\sinh }^2(x)=1$. So, the function simplifies to just $1$.

Differentiate the Constant Function

The derivative of a constant is always zero. Thus, the derivative of $1$ is $0$.

Graph the Function and Its Derivative

The graph of the function ${\cosh }^2(x)-{\sinh }^2(x)$, which simplifies to $1$, is a horizontal line at $y=1$. The derivative, which is $0$, is a line on the x-axis at $y=0$.

Key concepts

Hyperbolic Functions

Hyperbolic functions are analogues of traditional trigonometric functions but are based on hyperbolas instead of circles. Key hyperbolic functions include the hyperbolic sine ($\sinh (x)$) and hyperbolic cosine ($\cosh (x)$). You may think of them as important in calculus and engineering, offering unique properties for mathematical modeling.
These functions are defined as follows:

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

The interesting feature of hyperbolic functions is their identity relations, similar to trigonometric identities. One crucial identity is:
$${\cosh }^2(x)-{\sinh }^2(x)=1$$This identity was used in the original exercise to simplify the expression ${\cosh }^2(x)-{\sinh }^2(x)$ to 1. Understanding these identities helps streamline complex problems, just like simplifying trigonometry equations.

Graphing Functions

When graphing functions, visualizing their behavior across intervals is vital. Hyperbolic functions, like facts about their graphs, have unique characteristics. For instance, the graph of $\cosh (x)$ resembles a parabola, it's always positive and grows exponentially on either side of the y-axis.
In contrast, $\sinh (x)$ is an odd function, meaning its graph is symmetric about the origin. Visualizations of these functions allow us to:

• Comprehend growth and decay rates
• Recognize symmetry and periodicity
• Gain better insight into function behavior

For the problem at hand, the given expression simplifies to the constant 1, producing a flat horizontal line in a graph. Its derivative, equaling zero, appears as a line along the x-axis at $y=0$. Graphing gives a concrete illustration of how the expressions behave, reinforcing understanding.

Differentiation Process

Differentiation is a fundamental process in calculus, used to find the rate at which a function changes at any point. For hyperbolic functions, differentiation processes are similar to those of exponential and trigonometric functions.
The derivative of $\sinh (x)$ is $\cosh (x)$, and vice versa, the derivative of $\cosh (x)$ is $\sinh (x)$. In the exercise, since the function simplifies to a constant (1), we need to apply only a basic rule of differentiation: the derivative of any constant is zero.
This results in a steady horizontal graph line of the original function. Knowing these rules makes tackling such problems easier, as you can focus on simplifying the expression first before applying derivatives. Recognizing when and how to apply these rules efficiently is a cornerstone of mastering calculus.