Problem 46 - Calculus Single Variable

Question

Differentiate the given expression with respect to \(x\). $$ \operatorname{coth}\left(x^{2}\right) $$

Step-by-Step Solution

Answer

The derivative is \(-2x \operatorname{csch}^2(x^2)\).

Step 1: Identify the Function and Apply Chain Rule

The given function is \( \operatorname{coth}(x^2) \). Since it's a composite function, we will apply the chain rule. The outer function, \( \operatorname{coth}(u) \), requires the derivative \( \frac{d}{du} \left[ \operatorname{coth}(u) \right] = -\operatorname{csch}^2(u) \). For the inner function, \( u = x^2 \), the derivative is \( \frac{du}{dx} = 2x \). Therefore, the chain rule tells us that: \[ \frac{d}{dx} \left[ \operatorname{coth}(x^2) \right] = \frac{d}{du} \left[ \operatorname{coth}(u) \right] \times \frac{du}{dx}. \]

Step 2: Differentiate the Inner Function

Differentiate the inner function \( u = x^2 \) with respect to \( x \). This gives us: \[ \frac{du}{dx} = 2x. \]

Step 3: Differentiate the Outer Function

The derivative of \( \operatorname{coth}(u) \) with respect to \( u \) is \( \frac{d}{du} \left[ \operatorname{coth}(u) \right] = -\operatorname{csch}^2(u) \).

Step 4: Apply the Chain Rule

Combine the results from Steps 2 and 3 using the chain rule: \[ \frac{d}{dx} \left[ \operatorname{coth}(x^2) \right] = -\operatorname{csch}^2(x^2) \cdot 2x. \] This simplifies to: \[ -2x \operatorname{csch}^2(x^2). \]

Step 5: Write the Final Answer

Thus, the derivative of \( \operatorname{coth}(x^2) \) with respect to \( x \) is: \[ -2x \operatorname{csch}^2(x^2). \]

Key Concepts

Chain Rule

In calculus, the chain rule is a fundamental technique used to compute the derivative of a composite function. A composite function means you have one function nested inside another, like our example, \( \operatorname{coth}(x^2) \). Here's a simple way to understand the process:

• First, identify the outer function, which is \( \operatorname{coth}(u) \) in this case, where \( u = x^2 \).
• Take the derivative of the outer function with respect to its inner variable \( u \).


• Next, find the derivative of the inner function with respect to \( x \).


• Finally, multiply these derivatives together to get the result.

The chain rule effectively helps you "chain" together the derivatives of each function layer within the composition, ensuring every part of the nested function is correctly differentiated.

Hyperbolic Functions

Hyperbolic functions are counterparts to the standard trigonometric functions but based on hyperbolas rather than circles.These functions play a vital role in various fields such as engineering, physics, and calculus. In the context of our differentiation problem, we are focusing on the hyperbolic cotangent function, noted as \( \operatorname{coth}(x) \).The hyperbolic cotangent function is defined as:\[ \operatorname{coth}(x) = \frac{\cosh(x)}{\sinh(x)} \] where:

• \( \cosh(x) \) is the hyperbolic cosine and is defined as \( \cosh(x) = \frac{e^x + e^{-x}}{2} \).


• \( \sinh(x) \) is the hyperbolic sine and is defined as \( \sinh(x) = \frac{e^x - e^{-x}}{2} \).

The derivative of \( \operatorname{coth}(x) \) plays a crucial part in the differentiation process, as it simplifies to:\[ \frac{d}{dx}[\operatorname{coth}(x)] = -\operatorname{csch}^2(x) \]Thus, recognizing and using hyperbolic functions simplifies solving complex derivative problems like the one we have.

Derivative Computation

When dealing directly with the computation of derivatives, it's crucial to break down the function into more manageable parts.For our exercise, we are computing the derivative of \( \operatorname{coth}(x^2) \) with respect to \( x \). Here’s what each step involves:

• Identify the inner function \( u = x^2 \) and differentiate, resulting in \( \frac{du}{dx} = 2x \).


• Recognize the outer function \( \operatorname{coth}(u) \) and find its derivative using the formula for the derivative of a hyperbolic cotangent, \( \frac{d}{du} [\operatorname{coth}(u)] = -\operatorname{csch}^2(u) \).


• Combine these results by multiplying the derivative of the outer function by the derivative of the inner function, implementing the chain rule: \[ \frac{d}{dx}[\operatorname{coth}(x^2)] = -2x \operatorname{csch}^2(x^2) \]

With these steps, you can navigate the derivative computation effectively and understand the intricacies involved in finding derivatives of composite and hyperbolic functions.