Problem 389 - 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 of \( \cosh^2(x) + \sinh^2(x) \) is \( 2\sinh(2x) \).

Step 1: Recall Hyperbolic Identity

The expression given is \( \cosh^2(x) + \sinh^2(x) \). Recall the identity for hyperbolic functions: \( \cosh^2(x) - \sinh^2(x) = 1 \). Using this, we can rewrite the given expression.

Step 2: Simplify the Expression

Using the identity, we can write: \( \cosh^2(x) + \sinh^2(x) = 2\cosh^2(x) - 1 \). Thus, the expression simplifies to \( 2\cosh^2(x) - 1 \).

Step 3: Find the Derivative

Differentiate the simplified expression \( 2\cosh^2(x) - 1 \). Use the chain rule for \( \cosh(x) \): \( \frac{d}{dx} \cosh(x) = \sinh(x) \). The derivative is \( 2 \times 2\cosh(x)\sinh(x) = 4\cosh(x)\sinh(x) \).

Step 4: Verify the Derivative

Verify the derivative by recognizing that \( \cosh(x)\sinh(x) = \frac{1}{2}\sinh(2x) \). Therefore, the derivative \( 4\cosh(x)\sinh(x) \) can also be expressed as \( 2\sinh(2x) \).

Step 5: Graph the Original and Derived Functions

Graph the original function \( \cosh^2(x) + \sinh^2(x) \) and its derivative \( 2\sinh(2x) \). Ensure the shape of the derivative is a double-frequency sine wave, reflecting changes in the original graph.

Key Concepts

Understanding Hyperbolic Functions

Hyperbolic functions are similar to trigonometric functions, but they are based on hyperbolas instead of circles. They include functions like hyperbolic sine \(\sinh(x)\) and hyperbolic cosine \(\cosh(x)\). These functions are vital in calculus and help solve various types of real-world problems, particularly in physics and engineering.

• Hyperbolic Sine (\(\sinh(x)\)): Defined as \(\sinh(x) = \frac{e^x - e^{-x}}{2}\), this function shares various properties with the sine function.
• Hyperbolic Cosine (\(\cosh(x)\)): Defined as \(\cosh(x) = \frac{e^x + e^{-x}}{2}\), it resembles the properties of cosine.

They follow specific identities, such as \(\cosh^2(x) - \sinh^2(x) = 1\). This identity is pivotal in simplifying and solving exercises involving hyperbolic functions.
Recognizing these identities allows us to simplify expressions or equations easily. For example, in this exercise, we used the hyperbolic identity to simplify \(\cosh^2(x) + \sinh^2(x) = 2\cosh^2(x) - 1\).

By understanding and using these identities, you can approach hyperbolic problems more effectively.

The Chain Rule in Calculus

The chain rule is a fundamental technique for finding the derivative of composite functions. When a function contains another function as its argument, the chain rule helps us determine the derivative of this composition.
Here's a simple way to understand it:

• Chain Rule Formula: If you have a composite function \(f(g(x))\), the derivative is \(f'(g(x))\cdot g'(x)\).
• This means you differentiate the outer function, keep the inner function as it is, and then multiply by the derivative of the inner function.

In the provided exercise, we applied the chain rule to find the derivative of \(2\cosh^2(x) - 1\). Notice how we found \(\frac{d}{dx} \cosh(x) = \sinh(x)\), and applied it to solve for the derivative as \(4\cosh(x)\sinh(x)\).
The chain rule simplifies complex differentiation, even when dealing with hyperbolic functions. It's crucial for solving many types of calculus problems, especially those involving nested functions.

Graphing Functions and Their Derivatives

Graphing is an excellent way to understand the behavior of functions and their derivatives. By visually analyzing a graph, you can track how the original function changes and how it relates to its derivative.
In this scenario, we graphed the original function \(\cosh^2(x) + \sinh^2(x)\) and its derivative \(2\sinh(2x)\). Here's what graphing reveals:

• The original function's curve shows the specific transformations and behavior linked to hyperbolic identities.
• The derivative graph, in this case, resembles a double-frequency sine wave, which reflects how the rate of change in the original function's graph repeats with twice the frequency.

Understanding these visual aspects allows us to confirm our calculations and see the relationship between the original function and its derivative. For example, in our exercise, graphing both helped ensure that \(2\sinh(2x)\) accurately depicted changes in the original hyperbolic function's behavior.
Being familiar with graphing and how it relates to derivatives enhances your intuition and insight into functional behavior, making it a vital part of calculus study.