Question

Find the derivatives of the given functions and graph along with the function to ensure your answer is correct.[T] \(\tanh \left(\sqrt{x^{2}+1}\right)\)

Short answer

The derivative is \( f'(x) = \frac{x \cdot \text{sech}^2(\sqrt{x^2 + 1})}{\sqrt{x^2 + 1}} \).

Step-by-step solution

Understand the Function

The original function given is \( f(x) = \tanh(\sqrt{x^2 + 1}) \). We need to find its derivative and graph it along with the original function. The function involves the hyperbolic tangent and a square root expression.

Identify the Chain Rule

Notice that the function is a composition of two functions: \( g(u) = \tanh(u) \) and \( u(x) = \sqrt{x^2 + 1} \). To differentiate \( f(x) \), we need to use the chain rule for composition of functions.

Differentiate the Outer Function

Differentiate the outer function, \( g(u) = \tanh(u) \), which is \( g'(u) = \text{sech}^2(u) \). The derivative of \( \tanh(u) \) with respect to \( u \) is \( \text{sech}^2(u) \).

Differentiate the Inner Function

Differentiate the inner function, \( u(x) = \sqrt{x^2 + 1} \), which becomes \( \frac{d}{dx}\sqrt{x^2 + 1} = \frac{x}{\sqrt{x^2 + 1}} \) using the derivative \( \frac{d}{dx} \sqrt{v} = \frac{1}{2\sqrt{v}} \cdot 2x \) for \( v = x^2 + 1 \).

Apply the Chain Rule

The derivative of \( f(x) = \tanh(\sqrt{x^2 + 1}) \) using the chain rule is \( f'(x) = g'(u(x)) \cdot u'(x) = \text{sech}^2(\sqrt{x^2 + 1}) \cdot \frac{x}{\sqrt{x^2 + 1}} \).

Simplify the Derivative

Simplify the expression for clarity: \[ f'(x) = \frac{x \cdot \text{sech}^2(\sqrt{x^2 + 1})}{\sqrt{x^2 + 1}} \]

Graph the Functions

Graph the original function \( \tanh(\sqrt{x^2 + 1}) \) and its derivative \( \frac{x \cdot \text{sech}^2(\sqrt{x^2 + 1})}{\sqrt{x^2 + 1}} \) to verify that the derivative correctly represents the slope at each point on the original function.

Key concepts

Chain Rule

The chain rule is a fundamental technique in calculus used to differentiate composite functions, i.e., functions made up of two or more smaller functions. In the given exercise, our target function is \( f(x) = \tanh(\sqrt{x^2 + 1}) \). To manage the differentiation, we recognize two components: \( g(u) = \tanh(u) \) and \( u(x) = \sqrt{x^2 + 1} \). You start by differentiating the outer function with respect to its inner component and then multiply it by the derivative of the inner function itself.
When applying the chain rule, follow these simple steps:

• Identify the outer function \( g(u) \) and inner function \( u(x) \).
• Differentiate the outer function first, using the inner function as its variable.
• Multiply this derivative by the derivative of the inner function.

In our case, the derivative computation results with the final expression:
\[ f'(x) = \frac{x \cdot \text{sech}^2(\sqrt{x^2 + 1})}{\sqrt{x^2 + 1}} \]
This simplifies to ensure that both function composition and differentiation are fully understood.

Hyperbolic Functions

Hyperbolic functions like \( \tanh(x) \), or hyperbolic tangent, are functions that are analogous to trigonometric functions but for hyperbolas. They have important properties and appear in contexts involving hyperbolic geometry and complex number calculations. In our exercise, \( \tanh(x) \) plays a key role. It's defined by the relationship:
\[ \tanh(x) = \frac{\sinh(x)}{\cosh(x)} \]
where \( \sinh(x) \) and \( \cosh(x) \) are the hyperbolic sine and cosine, respectively.
Understanding the derivative of \( \tanh(x) \) is crucial when applying rules like chain rule. The derivative is:

• \( \frac{d}{dx} \tanh(x) = \text{sech}^2(x) \)

In the solution, understanding how \( \tanh(u) \) transforms when space inside the function, like \( \sqrt{x^2 + 1} \), shifts is key to correctly applying differentiative operations.

Graphing Functions

Graphing functions is not only essential for visualizing mathematical phenomena but also for verifying solutions. After computing derivatives like in our exercise, graphing ensures that the solution aligns with the behavior of the function.
In this exercise, we graph both: the original function \( \tanh(\sqrt{x^2 + 1}) \) and its derivative \( \frac{x \cdot \text{sech}^2(\sqrt{x^2 + 1})}{\sqrt{x^2 + 1}} \).
Here's why graphing is integral:

• It offers a visual check for differentiability and correctness.
• Displays function behaviors like increases or decreases.
• Helps comprehend how slopes change across different parts of the graph.

Use graphing tools or software to see how angle or curve variations represent the derivative's role. This tangible visualization helps in making complex ideas like function transformation more accessible and applicable.