Problem 26 - Calculus Early Transcendentals

Question

Compute \(d y / d x\) for the following functions. \(y=\sqrt{\operatorname{coth} 3 x}\)

Step-by-Step Solution

Answer

The derivative of the function \(y = \sqrt{\operatorname{coth} 3x}\) with respect to \(x\) is \(\frac{dy}{dx} = -\frac{3}{2\sqrt{\operatorname{coth} 3x} (\operatorname{sinh}(3x))^2}\).

Step 1: Compute the derivative of \(f(u)\) with respect to \(u\)

Using the power rule, we have \(\frac{d}{du}\sqrt{u} = \frac{1}{2\sqrt{u}}\)

Step 2: Compute the derivative of \(g(x)\) with respect to \(x\)

We have \(g(x) = \operatorname{coth} 3x\), so let's first find the derivative of \(\operatorname{coth} u\) with respect to \(u\). Recall that \(\operatorname{coth} u = \frac{\operatorname{cosh} u}{\operatorname{sinh} u}\). Using the quotient rule, we get \(\frac{d}{du} \operatorname{coth} u = \frac{ \operatorname{sinh} u(\operatorname{cosh} u)' - \operatorname{cosh} u(\operatorname{sinh} u)'}{(\operatorname{sinh} u)^2} = \frac{\operatorname{sinh} u( \operatorname{sinh} u ) - \operatorname{cosh} u( \operatorname{cosh} u)}{(\operatorname{sinh} u)^2} = \frac{(\operatorname{cosh} u)^2 - (\operatorname{sinh} u)^2}{(\operatorname{sinh} u)^2}\) Using the identity \(\operatorname{cosh}^2 u - \operatorname{sinh}^2 u = 1\), we then have \(\frac{d}{du} \operatorname{coth} u = -\frac{1}{(\operatorname{sinh} u)^2}\) Now, we can find the derivative of \(g(x)\) with respect to \(x\) using the chain rule: \(\frac{d}{dx}\operatorname{coth} 3x = -\frac{1}{(\operatorname{sinh} (3x))^2} (3) = -\frac{3}{(\operatorname{sinh} (3x))^2}\)

Step 3: Apply the chain rule

Finally, using the chain rule to find the derivative of \(y\) with respect to \(x\), we get \(\frac{dy}{dx} = \frac{d}{dx}\sqrt{\operatorname{coth} 3x} = \frac{1}{2\sqrt{\operatorname{coth} 3x}}\cdot\left(-\frac{3}{(\operatorname{sinh} (3x))^2}\right) = -\frac{3}{2\sqrt{\operatorname{coth} 3x} (\operatorname{sinh}(3x))^2}\) So, the derivative of the given function with respect to \(x\) is: \(\frac{dy}{dx} = -\frac{3}{2\sqrt{\operatorname{coth} 3x} (\operatorname{sinh}(3x))^2}\)

Key Concepts

Chain Rule

The chain rule is a fundamental tool in calculus for finding the derivative of composite functions, which are functions whose inputs are also functions. It allows you to "link" the derivatives of the inner and outer functions together for this purpose. When dealing with composite functions of the form \(f(g(x))\), the chain rule states that the derivative \(\frac{d}{dx}f(g(x))\) is the derivative of the outer function \(f\) evaluated at the inner function \(g(x)\), multiplied by the derivative of the inner function \(g(x)\).

• Mathematically expressed, it is: \( \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) \).
• This rule is crucial when dealing with nested functions, as it allows us to compute the outer function's contribution and the inner function's rate of change separately.

In the provided exercise, the chain rule is used to differentiate \( \sqrt{\operatorname{coth} 3x} \), where \(f(u) = \sqrt{u}\) and \(g(x) = \operatorname{coth} 3x\). The derivative calculation involves taking the derivative of \(f\) with respect to \(u\) and then multiplying by the derivative of \(g\) with respect to \(x\), effectively "chaining" them together.

Hyperbolic Functions

Hyperbolic functions are analogs of trigonometric functions that are based on hyperbolas instead of circles. They are important in various mathematical applications, particularly in calculus and complex analysis. The main hyperbolic functions are the hyperbolic sine \(\operatorname{sinh} x\) and hyperbolic cosine \(\operatorname{cosh} x\), from which other hyperbolic functions are derived.

• \(\operatorname{coth}(x)\), or hyperbolic cotangent, is one such derived function, defined as \( \operatorname{coth}(x) = \frac{\operatorname{cosh} x}{\operatorname{sinh} x} \).
• The identity \(\operatorname{cosh}^2 u - \operatorname{sinh}^2 u = 1\) is particularly useful, similar to the Pythagorean identity in trigonometry.

In the provided example, the derivative of \(g(x) = \operatorname{coth}(3x)\) is derived using this identity. This involves expressing \(\operatorname{coth}(u)\) in terms of \(\operatorname{sinh}(u)\) and \(\operatorname{cosh}(u)\), utilizing their derivatives and applying simplification based on the identity. In calculus, knowing how to differentiate hyperbolic functions is often vital to solving more complex problems.

Quotient Rule

The quotient rule is a method for finding the derivative of a function that is the ratio of two differentiable functions. It is particularly useful when the standard rules for derivatives, such as the product rule, do not easily apply. For two functions \(u(x)\) and \(v(x)\), if we have a function \(h(x) = \frac{u(x)}{v(x)}\), the derivative \(h'(x)\) can be found using the quotient rule.

• The formula for the quotient rule is: \[ \frac{d}{dx} \left( \frac{u(x)}{v(x)} \right) = \frac{v(x)u'(x) - u(x)v'(x)}{[v(x)]^2} \].
• This formula helps avoid the messiness of manually working through derivatives of such ratios.

In the given solution, the quotient rule is applied to differentiate \(\operatorname{coth}(u)\), represented as \(\frac{\operatorname{cosh}(u)}{\operatorname{sinh}(u)}\). This finds the derivative by carefully combining the individual components, revealing a smooth process for handling the more complex derivative of the ratio. The quotient rule simplifies this step, ensuring the mathematical process is both efficient and clear.