Problem 13 - Essential Calculus Early Transcendentals

Question

$$\frac{d}{d x}\left(\cot ^{-1} x\right)=-\frac{1}{1+x^{2}}$$

Step-by-Step Solution

Answer

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

Step 1: Recognize the Inverse Function

The function given is \(\cot^{-1}(x)\), which is the inverse cotangent function, also known as the arccotangent.

Step 2: Differentiate the Inverse Function

The derivative of \(\cot^{-1}(x)\) is a standard result. The derivative is \(-\frac{1}{1+x^2}\), which can be derived using implicit differentiation or by knowing the standard derivative formulas for inverse trigonometric functions.

Step 3: Confirm the Formula

Confirm that the derivative of \(\cot^{-1}(x)\) indeed matches the given expression. We find that \(\frac{d}{dx}(\cot^{-1}(x)) = -\frac{1}{1+x^2}\), which is correct based on inverse trigonometric derivative formulas.

Key Concepts

Understanding Derivative of Inverse Functions

Inverse functions are fascinating in calculus because they allow us to reverse the input-output relations of functions. When we talk about derivatives in this context, it's often about finding how quickly these inverse functions change with respect to changes in the input. The derivative of an inverse function like \(f^{-1}(x)\) can be found using the relation:

• If \(y = f(x)\), then \(x = f^{-1}(y)\).
• The derivative \[\frac{d}{dx}(f^{-1}(x)) = \frac{1}{f'(f^{-1}(x))}\].

By applying this to inverse trigonometric functions, specifically the arccotangent \(\cot^{-1}(x)\), we use known derivatives of their respective trigonometric identities. The cotangent function's behavior complements and affects the derivative of its inverse. Hence, the result \(-\frac{1}{1+x^2}\) gives a compact way to describe the rate of change for \(\cot^{-1}(x)\). Calculating these derivatives demands familiarity with trigonometric identities and implicit differentiation techniques.

Exploring the Cotangent Function

The cotangent function, denoted by \(\cot(x)\), is one of the fundamental trigonometric functions. It can be expressed as the ratio of the cosine to the sine function: \(\cot(x) = \frac{\cos(x)}{\sin(x)}\). This function is important in both pure and applied mathematics, often appearing in geometry and wave analyses. In terms of its graph, it passes through the origin and has vertical asymptotes at every integer multiple of \(\pi\).

The significance of the cotangent in the context of inverse functions is that \(\cot^{-1}(x)\), or arccotangent, assigns an angle to a ratio, reflecting how cotangent values vary with angles. To differentiate inverse trigonometric functions like this, we must consider the behavior and limits of the original trigonometric function. It's crucial to note that \(\cot^{-1}(x)\) is defined for all real numbers and its range is typically \(0 < y < \pi\).

Understanding these properties helps in grasping how derivatives of inverse trigonometric functions are derived, particularly the role played by various trigonometric identities and relationships.

Basics of Differential Calculus

Differential calculus is a branch of mathematics focused on how things change. It introduces the concept of the derivative, which represents the rate of change of a function. Most vividly, derivatives tell us how a dynamic system evolves, such as how velocity affects position in physics. Central to differential calculus is understanding how to find these derivatives through techniques like the power rule, product rule, and chain rule.

• The derivative provides a formula to compute tangents to curves, giving insight into the curve’s behavior at any point.
• Essential calculus operations like differentiation require finding the limit of the average rate of change as it approaches a specific point.

In our particular example of \(\frac{d}{d x}\left(\cot^{-1} x\right) = -\frac{1}{1+x^{2}}\), we see differentiation applied within inverse trigonometric functions, demonstrating real-world use in mathematical modeling and analysis. Differential calculus, thus, serves as an indispensable tool for studying the intricate nuances of change in various scientific fields.