Problem 386 - 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. $$\sinh \left(x^{2}\right)$$

Step-by-Step Solution

Answer

Derivative: \(2x \cdot \cosh(x^2)\).

Step 1: Understand the Function

We are asked to find the derivative of the function \( f(x) = \sinh(x^2) \). The function involves the hyperbolic sine function, \( \sinh(u) \), where \( u = x^2 \).

Step 2: Recall the Derivative of Hyperbolic Sine

The derivative of the hyperbolic sine function \( \sinh(u) \) with respect to \( u \) is \( \cosh(u) \). Therefore, \( \frac{d}{du}[\sinh(u)] = \cosh(u) \).

Step 3: Use the Chain Rule

Since \( u = x^2 \), we'll use the chain rule to find \( \frac{d}{dx}[\sinh(x^2)] \). The chain rule states that \( \frac{d}{dx}[f(u)] = f'(u) \cdot \frac{du}{dx} \). In this case, \( f(u) = \sinh(u) \) and \( u = x^2 \).

Step 4: Derivative of the Inner Function

Calculate \( \frac{du}{dx} \) where \( u = x^2 \). The derivative \( \frac{d}{dx}[x^2] = 2x \).

Step 5: Apply the Chain Rule

Using the chain rule, \( \frac{d}{dx}[\sinh(x^2)] = \cosh(x^2) \cdot 2x \). This gives us the derivative as \( 2x \cdot \cosh(x^2) \).

Step 6: Graph the Function and Its Derivative

To confirm the result, graph \( f(x) = \sinh(x^2) \) and its derivative \( f'(x) = 2x \cdot \cosh(x^2) \). Check that the graph of the derivative correctly represents the slope behavior of the original function.

Key Concepts

Hyperbolic Functions

Hyperbolic functions are similar to trigonometric functions, but they are based on hyperbolas rather than circles. Two of the most common hyperbolic functions are the hyperbolic sine, \( \sinh(x) \), and the hyperbolic cosine, \( \cosh(x) \). These functions have unique properties and are often used in various fields such as engineering and physics.
For the hyperbolic sine function, \( \sinh(x) \), the formula is given by:

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

Its derivative with respect to \( x \) is the hyperbolic cosine function, \( \cosh(x) \), which is defined as:

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

This means that when we differentiate \( \sinh(x) \), we get \( \cosh(x) \).
This characteristic is particularly useful as it mirrors the relationship between sine and cosine functions in trigonometry, while still being distinct. These hyperbolic functions are especially useful in calculus when modeling real-world phenomena involving hyperbolic shapes.

Chain Rule

In calculus, the chain rule is a fundamental technique used for finding the derivative of composite functions. A composite function is one where a function is applied to another function, such as \( f(g(x)) \).
The chain rule allows us to differentiate these types of functions efficiently. It states that if we have a function \( f(u) \) where \( u = g(x) \), the derivative with respect to \( x \) is given by:

• \( \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) \)

This means we first differentiate the outer function, \( f \), with respect to \( u \), and then multiply it by the derivative of the inner function, \( g \), with respect to \( x \).
Let's apply the chain rule to our original function \( \sinh(x^2) \).
The derivative of \( \sinh(x^2) \) is calculated by first taking the derivative of \( \sinh(u) \), which is \( \cosh(u) \), and then multiplying it by the derivative of \( x^2 \), which is \( 2x \). Thus, the derivative is \( 2x \cdot \cosh(x^2) \).
The chain rule ensures that we account for the rate of change of both the outer and inner functions, giving us an accurate slope of the function. It is a powerful tool that simplifies solving complex differentiation tasks.

Graphing Derivatives

Graphing derivatives is a visual method to verify and understand the behavior of functions. Graphing the original function alongside its derivative can help us see how the derivative represents the slope of the original function at any given point.
When graphing \( f(x) = \sinh(x^2) \) and its derivative \( f'(x) = 2x \cdot \cosh(x^2) \), consider the following:

• The graph of \( f(x) = \sinh(x^2) \) will typically exhibit a steep curve because \( \sinh \) grows rapidly as \( x^2 \) increases.
• The derivative \( f'(x) \), graphically displays the slope or rate of change of \( f(x) \).
• A positive value in the derivative graph indicates a region where the original function is increasing, while a negative value suggests it is decreasing.

By graphing both, you can confirm that \( f'(x) = 2x \cdot \cosh(x^2) \) is correct if the behavior of the graph of \( f(x) \) matches what the derivative indicates.
This process is invaluable in calculus, helping students visually grasp concepts such as maxima, minima, and points of inflection.