Question

Find the derivatives of the given functions and graph along with the function to ensure your answer is correct.[T] \(\cosh (3 x+1)\)

Short answer

The derivative is \(3\sinh(3x+1)\).

Step-by-step solution

Review Hyperbolic Cosine Function

We need to derive the hyperbolic cosine function. Recall that the derivative of the hyperbolic cosine function \( \cosh(x) \) is \( \sinh(x) \). Keeping this in mind will be crucial for finding the derivative of the given function \( \cosh(3x+1) \).

Identify the Inner Function

Analyze the argument \(3x+1\) within the hyperbolic cosine function. Notice the presence of an inner function, \(u = 3x + 1\). We will use the chain rule, which requires us to differentiate both the outer function and its inner function.

Apply the Chain Rule

The chain rule for derivatives states \( \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) \). Set \( f(u) = \cosh(u) \) and \( g(x) = 3x + 1 \). Differentiate these to find \( f'(u) = \sinh(u) \) and \( g'(x) = 3 \). Apply the rule: \( \frac{d}{dx}[\cosh(3x+1)] = \sinh(3x+1) \cdot 3 \).

Simplify the Derivative

Combine the results from the previous step. The derivative of the function \( \cosh(3x + 1) \) is \( 3\sinh(3x + 1) \).

Graph the Functions

Create graphs for both \( y = \cosh(3x+1) \) and its derivative \( y = 3\sinh(3x+1) \). Use graphing software or a graphing calculator to visualize these functions. The graph of the derivative should reflect changes in the slope of \( y = \cosh(3x+1) \).

Key concepts

Hyperbolic Functions

Hyperbolic functions are mathematical functions that have similarities with trigonometric functions but are based on hyperbolas instead of circles. These functions are significant in various fields such as calculus and engineering.
The two most common hyperbolic functions are the hyperbolic sine (\( \sinh(x) \)) and the hyperbolic cosine (\( \cosh(x) \)).

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

In our task, the function we are dealing with is a hyperbolic cosine function, \( \cosh(3x+1) \). It behaves similarly to regular cosine but exhibits exponential growth and decay patterns. Understanding the basic derivatives of hyperbolic functions is crucial, as in this case where the derivative of \( \cosh(x) \) is \( \sinh(x) \).

Derivatives

Derivatives measure how a function changes as its input changes. They are fundamental in calculus for understanding concepts such as slope, rates of change, and optimization. When we derive a function, we are essentially calculating its rate of change.
For hyperbolic functions like \( \cosh(x) \), knowing its derivative is critical. The derivative of \( \cosh(x) \) is \( \sinh(x) \). Applying this knowledge, when we look at the function \( \cosh(3x+1) \), we must consider both the outer function (hyperbolic cosine) and the inside argument (3x+1). This is where derivatives become slightly more complex, and techniques like the chain rule come into play.

Chain Rule

The chain rule is a technique in calculus used for computing the derivative of the composition of two or more functions. It's particularly useful when dealing with composite functions where you have a function inside another function.
Here's how the chain rule works: If you have a function \( y = f(g(x)) \), where \( g(x) \) is the inner function and \( f(u) \) is the outer function, then the derivative of \( y \) with respect to \( x \) is given by:\[ \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) \]In the case of our function \( \cosh(3x+1) \), the chain rule allows us to find its derivative by first deriving the outer function \( \cosh(u) \) to get \( \sinh(u) \), and then deriving the inner function \( 3x+1 \) to obtain 3. The final step is to multiply these derivatives, which results in \( 3\sinh(3x+1) \). This process demonstrates how the chain rule simplifies the differentiation of composite functions. By breaking it down into manageable steps, it becomes easier to handle, even for more complicated functions.