Problem 43 - Calculus Single Variable

Question

Differentiate the given expression with respect to \(x\). $$ \tanh (\tan (x)) $$

Step-by-Step Solution

Answer

The derivative is \( \text{sech}^2(\tan(x)) \cdot \sec^2(x) \).

Step 1: Understand the Nested Functions

The expression \( \tanh(\tan(x)) \) is a composition of two functions: \( \tanh(u) \) where \( u = \tan(x) \). Our goal is to find the derivative of this composite function with respect to \( x \).

Step 2: Apply the Chain Rule

To differentiate \( \tanh(\tan(x)) \), use the chain rule: \( \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) \). Here, \( f(u) = \tanh(u) \) and \( g(x) = \tan(x) \).

Step 3: Differentiate \( \tanh(u) \)

The derivative of \( \tanh(u) \) with respect to \( u \) is \( \text{sech}^2(u) \), where \( \text{sech}(u) = \frac{1}{\cosh(u)} \).

Step 4: Differentiate \( \tan(x) \)

The derivative of \( \tan(x) \) with respect to \( x \) is \( \sec^2(x) \).

Step 5: Combine Using the Chain Rule

Multiply the derivatives from Step 3 and Step 4: \[ \frac{d}{dx} \tanh(\tan(x)) = \text{sech}^2(\tan(x)) \cdot \sec^2(x). \]

Step 6: Conclusion: Final Derivative

The derivative of \( \tanh(\tan(x)) \) with respect to \( x \) is \( \text{sech}^2(\tan(x)) \cdot \sec^2(x) \).

Key Concepts

Chain Rule

The chain rule is a fundamental concept in differentiation used to find the derivative of composite functions. A composite function is a function inside another function, like the expression in our exercise, \( \tanh(\tan(x)) \). The chain rule is particularly useful when functions are nested within one another. Here’s how it works: given two functions \( f \) and \( g \), if you have a composite function \( f(g(x)) \), the chain rule states that its derivative is given by:

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

In simple terms, you differentiate the outer function, \( f \), as if the inner function was just a variable, and then multiply it by the derivative of the inner function, \( g \). In our problem, this means differentiating the hyperbolic function \( \tanh \), treating its input \( \tan(x) \) like a single unit, and then multiplying by \( \tan(x)'s \) derivative.
Understanding the chain rule is essential because it lets us tackle more complex functions built from simpler ones, expanding our ability to solve various calculus problems effectively.

Hyperbolic Functions

Hyperbolic functions are similar in form to trigonometric functions, but they arise from hyperbolas instead of circles. One such hyperbolic function is the hyperbolic tangent, denoted as \( \tanh(x) \). This function is defined as:

• \( \tanh(x) = \frac{\sinh(x)}{\cosh(x)} \)

Where \( \sinh(x) \) is the hyperbolic sine and \( \cosh(x) \) is the hyperbolic cosine, their definitions are:

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

For differentiation, \( \tanh(x) \) has a simple derivative:

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

Where \( \text{sech}(u) = \frac{1}{\cosh(u)} \) is the hyperbolic secant function, uniquely derived from hyperbolic identities. In the original exercise, we used the derivative of \( \tanh \) to apply the chain rule when differentiating \( \tanh(\tan(x)) \).
Incorporating hyperbolic functions into differentiation problems expands the scope of functions you can work with, showcasing the rich behaviors hyperbolic functions exhibit.

Trigonometric Functions

Trigonometric functions arise from the geometry of right triangles and the unit circle. In our exercise, we deal with the tangent function, \( \tan(x) \), which is one of these fundamental trigonometric functions. The tangent of an angle \( x \) is defined as:

• \( \tan(x) = \frac{\sin(x)}{\cos(x)} \)

For differentiation purposes, the derivative of \( \tan(x) \) is well-known:

• \( \frac{d}{dx} \tan(x) = \sec^2(x) \)

Where \( \sec(x) = \frac{1}{\cos(x)} \) represents the secant function. This derivative arises from the properties of sine and cosine derivatives. Knowing how to differentiate \( \tan(x) \) is essential when using the chain rule, as seen in our example. Since \( \tan(x) \) is part of the composite function, we needed this derivative to find the overall derivative of \( \tanh(\tan(x)) \).
Grasping the derivatives of trigonometric functions allows you to solve a wider array of calculus problems, as these functions frequently appear in mathematical and real-world contexts.