Problem 62 - Calculus for Biology and Medicine

Question

Use the quotient rule to show that $$ \frac{d}{d x} \cot x=-\csc ^{2} x $$

Step-by-Step Solution

Answer

The derivative of \( \cot x \) is \( -\csc^2 x \).

Step 1: Write the Definition of Cotangent

The cotangent function is defined as the reciprocal of the tangent function: \( \cot x = \frac{1}{\tan x} \). Thus, we can express it in terms of sine and cosine: \( \cot x = \frac{\cos x}{\sin x} \).

Step 2: Apply the Quotient Rule

To differentiate \( \cot x = \frac{\cos x}{\sin x} \), use the quotient rule: if \( u(x) = \cos x \) and \( v(x) = \sin x \), then \( \frac{d}{dx}\left( \frac{u}{v} \right) = \frac{v(x)u'(x) - u(x)v'(x)}{(v(x))^2} \).

Step 3: Find the Derivatives of \( u \) and \( v \)

Differentiate \( u = \cos x \) to get \( u' = -\sin x \). Differentiate \( v = \sin x \) to get \( v' = \cos x \).

Step 4: Substitute Derivatives into the Quotient Rule

Substitute \( u = \cos x \), \( u' = -\sin x \), \( v = \sin x \), and \( v' = \cos x \) into the quotient rule formula: \( \frac{v(x)u'(x) - u(x)v'(x)}{(v(x))^2} = \frac{\sin x (-\sin x) - \cos x (\cos x)}{(\sin x)^2} \).

Step 5: Simplify the Expression

Simplify \( \frac{\sin x (-\sin x) - \cos x (\cos x)}{(\sin x)^2} \) to obtain \( \frac{-\sin^2 x - \cos^2 x}{\sin^2 x} \). Using the Pythagorean identity \( \sin^2 x + \cos^2 x = 1 \), this becomes \( \frac{-1}{\sin^2 x} \).

Step 6: Express in Terms of \( \csc \)

Recognize that \( \csc x \) is the reciprocal of \( \sin x \), i.e., \( \csc x = \frac{1}{\sin x} \). Therefore, \( \frac{-1}{\sin^2 x} = -\csc^2 x \). Thus, \( \frac{d}{d x} \cot x = -\csc^2 x \).

Key Concepts

Cotangent Function

The cotangent function, abbreviated as cot, is a trigonometric function. It is related to both the sine and cosine functions. The cotangent function is defined as the reciprocal of the tangent function. So, mathematically, we write this as \( \cot x = \frac{1}{\tan x} \). Given that tangent is \( \tan x = \frac{\sin x}{\cos x} \), the cotangent, in turn, can be expressed as \( \cot x = \frac{\cos x}{\sin x} \).
This expression shows the cotangent in terms of sine and cosine which helps to understand its behavior and relationship within the trigonometric circle. In simple terms, the cotangent function mainly deals with the ratio of an angle's adjacent side over its opposite side in a right-angled triangle. Understanding cotangent is crucial for mastering trigonometry as it frequently appears in calculus, particularly in derivative-related problems.

Derivatives

In calculus, the concept of derivatives represents how a function changes as its input changes. It helps understand the rate of change and the behavior of the function. The derivative of a function at a point is the slope of the tangent line to the curve at that point.
To find the derivative of the cotangent function, we utilize the quotient rule, which is a method for finding the derivative of a quotient of two functions. If you have a function \( \frac{u}{v} \), where both \( u \) and \( v \) are functions of \( x \), the derivative is given by:

• First, find the derivative of the numerator, \( u \), and the denominator, \( v \).
• Apply the formula: \( \frac{v(x)u'(x) - u(x)v'(x)}{(v(x))^2} \).

This method allows us to systematically break down complex functions like \( \cot x = \frac{\cos x}{\sin x} \) making it easier to differentiate.
Practicing derivatives with different functions enhances the understanding of calculus concepts and prepares students for problems involving the rate of change and graphs of functions.

Pythagorean Identity

The Pythagorean identity is a fundamental relationship in trigonometry which stems from Pythagoras' Theorem. It relates the square of sine and cosine functions in such a way that it remains true for any angle. Precisely, the identity is \( \sin^2 x + \cos^2 x = 1 \). It is pivotal in trigonometry and calculus for simplifying expressions and solving equations.
In the derivative of the cotangent function, we make use of this identity during the simplification steps. Specifically, when simplifying expressions like \( -\sin^2 x - \cos^2 x \), the Pythagorean identity allows us to replace \( \sin^2 x + \cos^2 x \) with 1, thus simplifying the expression to \( -1 \).
Understanding and recognizing how to apply the Pythagorean identity can greatly ease calculations, making it an essential tool in both high school and advanced mathematics. Integrating this identity into problems often allows for simpler solutions and reveals deeper interconnections between trigonometric functions.