Question

Evaluate the derivative of the function at the indicated point. Use a graphing utility to verify your result. $$ y=\frac{1+\csc x}{1-\csc x} \quad\left(\frac{\pi}{6},-3\right) $$

Short answer

The derivative of the function \( y=\frac{1+\csc x}{1-\csc x} \) at the point \( \frac{\pi}{6},-3 \) is -3

Step-by-step solution

Simplify the Function

First, rewrite the given function in terms of \( \sin x \) \( y=\frac{1+\frac{1}{\sin x}}{1-\frac{1}{\sin x}} \)

Compute the Derivative

Next, we compute the derivative of this function using the quotient rule, which gives us \(y' = \frac{(1 - \frac{1}{\sin x})\cdot 0 - (1 + \frac{1}{\sin x})(-\cos x) \cdot (\sin x)^{-2}}{(1-\frac{1}{\sin x})^2}\). Simplifying yields \(y' = \frac{(\cos x) \cdot (\sin x)^{-2}}{(1-\frac{1}{\sin x})^2}\)

Evaluate the Derivative at the given Point

Substitute \( x= \frac{\pi}{6} \) into the derivative to evaluate the derivative at the given point, so we have \(y'(\frac{\pi}{6}) = \frac{(\cos \frac{\pi}{6}) \cdot (\sin \frac{\pi}{6})^{-2}}{(1-\frac{1}{\sin \frac{\pi}{6}})^2} = -3\).

Verify the result using a Graphing Utility

Finally, one can use any graphing utility to plot the function and its derivative to verify these analytical results visually. At \( x= \frac{\pi}{6} \), the slope of the function should be -3.

Key concepts

Graphing Utility Verification

A graphing utility is an indispensable tool for visually verifying the results of a calculus problem, especially when it involves derivatives and functions that are not straightforward. After you’ve performed an analytical solution to find derivatives, graphing the original function and its derivative side-by-side can provide a visual confirmation of the results.
For instance, when evaluating the derivative at a specific point, a graph can show you the slope of the tangent line at that point. By checking the derivatives graphically, one can verify computational results and gain a better intuitive understanding of the function's behavior. It is always recommended to use a graphing utility as an additional check after solving problems—seeing the problem and the solution graphically can sometimes offer insights that purely algebraic manipulations cannot.

Simplifying Expressions

Mathematical elegance often lies in the simplification of expressions, making complex problems more manageable and reducing the potential for errors in calculations. Simplifying an expression involves reducing it to its simplest form by performing all possible arithmetic operations, combining like terms, and eliminating complex fractions.
Simplification is not only aesthetically pleasing but also practical. In calculus, it minimizes the difficulty of taking derivatives or integrals and can reveal hidden relationships between variables. When you're preparing to differentiate a function, such as by the quotient rule, begin by simplifying the function as much as possible. This way, you'll avoid the pitfalls of unwieldy algebra and keep the focus on the calculus concepts at hand. Always look for opportunities to simplify; it's the foundation on which successful problem solving is built.