Problem 40 - University Calculus: Early Transcendentals

Question

Find the derivative of \(y\) with respect to the appropriate variable. $$y=\cot ^{-1} \frac{1}{x}-\tan ^{-1} x$$

Step-by-Step Solution

Answer

The derivative is \(-\frac{2}{1+x^2}\).

Step 1: Apply differentiation rules

Use power, product, quotient, or chain rule as appropriate.

Step 2: Simplify

The derivative is The derivative is \(-\frac{2}{1+x^2}\)..

Key Concepts

Understanding Derivatives

Derivatives play a crucial role in calculus. They help to understand how a function changes at any point in its domain. The derivative of a function gives us the slope of the tangent line to the function at any given point. It tells us whether a function is increasing or decreasing and can be applied to solve a range of real-world problems, from determining the velocity of a moving object to finding the optimal cost in economics. The derivative, often denoted as \(f'(x)\) or \(\frac{dy}{dx}\), is essentially the limit of the average rate of change of the function as the change in the input approaches zero. This concept is vital when dealing with functions, including those involving inverse trigonometric elements, as we will see in the subsequent discussion.

Inverse Trigonometric Functions

Inverse trigonometric functions are the inverse operations of regular trigonometric functions. They allow us to determine the angle when the trigonometric ratio is known, instead of finding the ratio itself for a given angle. The basic inverse trigonometric functions include \(\sin^{-1}(x)\), \(\cos^{-1}(x)\), and \(\tan^{-1}(x)\). Each of these functions has its own domain and range, necessary for the function to be invertible. For instance, the domain of \(\tan^{-1}(x)\) is all real numbers, and its range is \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\). When dealing with derivatives of inverse trigonometric functions, we recognize that they involve specific rules. For example, the derivative of \(\tan^{-1}(x)\) is \(\frac{1}{1+x^2}\), highlighting how these rules differ from those of standard trigonometric functions.

Differentiation Rules

Differentiation rules are formulas that simplify the process of finding derivatives. For functions involving multiple components or transformations like inverse trigonometric functions, these rules make differentiation manageable and systematic. In the problem provided, both the chain rule and the standard derivative of inverse trigonometric functions are used. The derivative of \(\cot^{-1}(u)\) involves both the basic derivative \(-\frac{1}{1+u^2}\) and the chain rule since \(u\) is a function of \(x\) itself. By applying these rules, we handle the compound structure efficiently. Differentiation rules allow us to decompose complex derivatives into simple calculations, ensuring that the process is straightforward and error-free.