Problem 38 - Calculus for Biology and Medicine

Question

Find the derivative with respect to the independent variable. $$ f(x)=\tan x \cot x $$

Step-by-Step Solution

Answer

The derivative of \( f(x)=\tan x \cot x \) is 0.

Step 1: Understand the Functions Involved

Identify the functions involved in the expression. Here, \( f(x) = \tan(x) \cot(x) \). The tangent function, \( \tan(x) \), is defined as \( \frac{\sin(x)}{\cos(x)} \) and the cotangent function, \( \cot(x) \), is defined as \( \frac{\cos(x)}{\sin(x)} \).

Step 2: Simplify the Expression

Simplify the expression \( \tan(x) \cot(x) \). Since \( \cot(x) = \frac{1}{\tan(x)} \), the product becomes \( \tan(x) \cdot \cot(x) = \tan(x) \cdot \frac{1}{\tan(x)} = 1 \).

Step 3: Find the Derivative

Differentiate the simplified expression with respect to \( x \). Since \( 1 \) is a constant, its derivative is \( 0 \).

Step 4: Write the Final Answer

Combine the results from the previous steps. The derivative of \( f(x) = \tan(x) \cot(x) \) is \( 0 \).

Key Concepts

Understanding Trigonometric Functions

The exercise begins by identifying the trigonometric functions involved, specifically the tangent and cotangent functions. These functions are fundamental in trigonometry, which studies the relationships between angles and sides of triangles. - The tangent function, denoted as \( \tan(x) \), is calculated by the ratio of the opposite side to the adjacent side in a right triangle, which can be expressed as \( \frac{\sin(x)}{\cos(x)} \). This means it is the sine of an angle divided by its cosine.- The cotangent function, represented as \( \cot(x) \), is essentially the reciprocal of the tangent function, and it is given by \( \frac{\cos(x)}{\sin(x)} \). When looking at these trig ratios, cotangent can also be seen as the adjacent side over the opposite side in a triangle.Understanding these definitions is crucial because they form the basis for trigonometric simplification and differentiation. When working with trigonometric derivatives, knowing these relationships enables you to simplify complex expressions, making differentiation more straightforward. It's important to become comfortable moving between various trigonometric identities to simplify and solve problems efficiently.

Simplification of Trigonometric Expressions

Simplification is a key step in calculus problems, especially when dealing with trigonometric functions. In this exercise, the expression to simplify is \( \tan(x) \cdot \cot(x) \). To simplify, we recognize that \( \cot(x) \) is the reciprocal of \( \tan(x) \). This property allows us to simplify the expression:

• Since \( \cot(x) = \frac{1}{\tan(x)} \), multiply it by \( \tan(x) \) to get \( \tan(x) \cdot \frac{1}{\tan(x)} \).
• The \( \tan(x) \) in the numerator and denominator cancel each other out, resulting in 1.
• Hence, the simplified expression is just 1.

Simplification not only makes solving the derivative easier but also reduces computational errors. Recognizing reciprocal relationships and applying them to simplify expressions is a highly useful skill in calculus problem-solving. It turns a potentially complicated differentiation problem into an almost trivial one, as seen here.

Approach to Calculus Problem Solving

Calculus problem solving often involves a structured approach to find derivatives or integrals. This exercise illustrates this beautifully through the following steps:1. **Identify and Understand Functions Involved:** Before jumping into differentiation, understand the components—here, \( \tan(x) \) and \( \cot(x) \).2. **Simplify Where Possible:** By simplifying \( \tan(x) \cdot \cot(x) \) to 1, you transform a potentially complex expression into a simple constant.3. **Compute the Derivative:** Once simplified, finding the derivative becomes straightforward. The derivative of a constant, like 1, is always 0. 4. **Verification:** Double-check each step to ensure accuracy, especially when simplifying and differentiating.In calculus, systematically breaking down problems into smaller, manageable steps can dramatically improve accuracy and efficiency. It prevents oversight and ensures that each part of the problem is well-understood, leading to a correct solution. By practicing such structured problem-solving, complex calculus tasks become more approachable and solvable.