Problem 28 - Thomas Calculus

Question

In Exercises \(21-42,\) find the derivative of \(y\) with respect to the appropriate variable. $$ y=\csc ^{-1} \frac{x}{2} $$

Step-by-Step Solution

Answer

The derivative of \(y = \csc^{-1}\left(\frac{x}{2}\right)\) is \(\frac{dy}{dx} = -\frac{1}{\sqrt{x^2-4}}\) for \(|x| \geq 2\).

Step 1: Identify the Function to Differentiate

The function we need to find the derivative of with respect to \(x\) is \(y = \csc^{-1}\left(\frac{x}{2}\right)\).

Step 2: Recall the Derivative Formula for the Inverse Cosecant

The derivative of \(\csc^{-1}(u)\) with respect to \(u\) is \(-\frac{1}{|u|\sqrt{u^2-1}}\). Here, \(u = \frac{x}{2}\), so we will need to use the chain rule to differentiate with respect to \(x\).

Step 3: Apply the Chain Rule

Using the chain rule, the derivative is given by \(\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}\). First, compute \(\frac{du}{dx}\) for \(u = \frac{x}{2}\), which is \(\frac{1}{2}\).

Step 4: Compute \(\frac{dy}{du}\)

Substitute \(u = \frac{x}{2}\) into the derivative formula for \(\csc^{-1}\), to get \(\frac{dy}{du} = -\frac{1}{|\frac{x}{2}|\sqrt{\left(\frac{x}{2}\right)^2-1}}\). This simplifies to \(-\frac{2}{|x|\sqrt{\frac{x^2}{4}-1}}\).

Step 5: Simplify and Combine the Derivatives

Combine the results from Steps 3 and 4: \[ \frac{dy}{dx} = \left(-\frac{2}{|x|\sqrt{\frac{x^2}{4}-1}}\right) \times \frac{1}{2} = -\frac{1}{|x|\sqrt{\frac{x^2}{4}-1}}. \] This simplifies to \(-\frac{1}{\sqrt{x^2-4}}\) if \(|x| \geq 2\).

Step 6: Confirm Domain Constraints

For the expression under the square root to be valid and for \(\csc^{-1}\) to be defined, \(|x| \geq 2\).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are special types of functions that reverse the effect of traditional trigonometric functions like sine, cosine, and tangent. Specifically, for inverse cosecant, denoted as \(\csc^{-1}(x)\), we look for an angle \(\theta\) such that \(\csc( heta) = x\). The use of inverse trigonometric functions is common in calculus due to their unique properties, particularly when dealing with integrals and derivatives. In this exercise, we focus on the inverse cosecant function, which is less common than others like inverse sine or inverse cosine.

Inverse functions can only exist on a restricted domain. It is crucial to remember for \(\csc^{-1}(x)\) that the value of \(x\) must be outside the interval \([-1, 1]\), since cosecant values are extreme (greater than 1 or less than -1). This will become especially important when we consider domain constraints later.

Chain Rule

One of the cornerstones of differentiation in calculus, the chain rule, allows us to find the derivative of composite functions. It expresses the derivative of \(y = f(g(x))\) as \((dy/dx = f'(g(x)) \cdot g'(x))\). In simpler terms, you differentiate the outer function and multiply it by the derivative of the inner function.

In our problem, \(y = \csc^{-1} \rac{x}{2}\), the inner function is \(u = \rac{x}{2}\), while the outer function is \(\csc^{-1}(u)\). This means we first differentiate the inverse cosecant with respect to \(u\), and then \(u\) with respect to \(x\). By applying the chain rule, we calculated \(dy/du\) and \(du/dx\), which when multiplied provided the derivative \(dy/dx\).

This technique helps break down complex derivatives into manageable parts, enhancing clarity and precision.

Differentiation Techniques

Differentiation techniques cover a wide variety of rules and strategies to find derivatives. For inverse trigonometric functions, like the one in this problem, there are specific derivative formulas available.

We needed the formula for \(\csc^{-1}(u)\), which is \(-\frac{1}{|u|\sqrt{u^2 - 1}}\). This formula helps us understand how fast the \(\csc^{-1}(u)\) function changes with \(u\). In combination with the chain rule, which we applied earlier, we differentiated the function efficiently.

Once you understand these formulas and techniques, transforming complex problems into simpler steps becomes feasible. By simplifying each part, like substituting \(u\) into the formula, the derivative is easily obtained.

Domain Constraints

Domain constraints refer to restrictions on the values that can be input into a function. These constraints ensure the function outputs valid results, avoiding undefined expressions. For \(\csc^{-1}(x)\), the domain constraint is that \(x\) must be \(|x| \\geq 2\). This is because cosecant, being the reciprocal of sine, only exists when its inputs result in non-zero sine values, which happens beyond the \([-1, 1]\) interval.

In our derivative solution, the constraint \(|x| = \\geq 2\) also affects the resulting expression. We need \(x^2 - 4 > 0\) to ensure the square root is defined. Calculating derivatives while respecting these domain constraints is vital; it ensures the derivative expression is meaningful and applicable under real-world scenarios.