Problem 49 - Calculus for Biology and Medicine

Question

In Problems \(1-58\), find the derivative with respect to the independent variable. $$ h(x)=\frac{3}{\tan (2 x)-x} $$

Step-by-Step Solution

Answer

The derivative is \(h'(x) = \frac{-6\sec^2(2x) + 3}{(\tan(2x) - x)^2}\)."

Step 1: Identify the Function and Its Components

The given function is expressed as a quotient: \(h(x) = \frac{3}{\tan(2x) - x}\). Here, the numerator is \(3\), and the denominator is \(\tan(2x) - x\). This forms a rational function where both elements are functions of \(x\).

Step 2: Apply the Quotient Rule

To differentiate a function of the form \(\frac{u}{v}\), use the quotient rule: \((\frac{u}{v})' = \frac{v \cdot u' - u \cdot v'}{v^2}\). For \(h(x)\), let \(u = 3\) and \(v = \tan(2x) - x\).

Step 3: Differentiate the Numerator and Denominator

Since \(u = 3\), its derivative \(u' = 0\). For \(v = \tan(2x) - x\), differentiate to get \(v' = \sec^2(2x) \cdot 2 - 1 = 2\sec^2(2x) - 1\).

Step 4: Substitute Derivatives into the Quotient Rule Formula

Substitute \(u' = 0\), \(u = 3\), \(v' = 2\sec^2(2x) - 1\), and \(v = \tan(2x) - x\) into the quotient rule formula: \[h'(x) = \frac{(\tan(2x) - x) \cdot 0 - 3 \cdot (2\sec^2(2x) - 1)}{(\tan(2x) - x)^2}\].

Step 5: Simplify the Expression

Simplify the expression from Step 4: \[h'(x) = \frac{-3(2\sec^2(2x) - 1)}{(\tan(2x) - x)^2} = \frac{-6\sec^2(2x) + 3}{(\tan(2x) - x)^2}\].

Key Concepts

Quotient Rule

The quotient rule is a fundamental tool in calculus used when differentiating functions that are expressed as quotients, meaning one function divided by another. This rule is specifically useful for functions where you have a numerator and a denominator that are both functions of the independent variable.

When you need to find the derivative of a function \(rac{u}{v}\), where both \(u\) and \(v\) are differentiable functions of \(x\), the quotient rule comes into play: \(\left( \frac{u}{v} \right)' = \frac{v \cdot u' - u \cdot v'}{v^2}\).

This formula provides the derivative of the quotient of two functions by

• Taking the derivative of the top function \(u\) and the bottom function \(v\),
• Multiplying the derivative of the top by the original bottom,
• Subtracting the product of the original top times the derivative of the bottom,
• And then dividing those results by the square of the bottom function.

In the example problem, the function is \(h(x) = \frac{3}{\tan(2x) - x}\), where the numerator \(u\) is simply \(3\) and the denominator \(v\) is \(\tan(2x)-x\). Understanding the quotient rule's steps, particularly the subtraction of products and the importance of \(v^2\) in the denominator, is crucial for solving complex calculus problems that involve fractions.

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. They are foundational in calculus, particularly when differentiating and integrating, because they often appear in scientific and engineering problems.

In calculus, some of the core trigonometric functions are \( \sin(x), \cos(x), \tan(x), \sec(x), \csc(x),\) and \(\cot(x)\). Each of these has specific derivatives important for calculus operations:

• The derivative of \(\tan(x)\) is \(\sec^2(x)\).
• The derivative of a transformed trigonometric function like \(\tan(2x)\) involves the chain rule, resulting in \(2\sec^2(2x)\).
• Simplifying these derivatives sometimes involves recognizing identities: e.g., \(\sec(x) = \frac{1}{\cos(x)}\).

In the exercise, the derivative of the term \(\tan(2x) - x\) in the denominator includes using the derivative \(\sec^2(2x) \) with the chain rule, which multiplies the result by the inner derivative (in this case, \(2\)). Understanding these derivatives and transformations is vital for applying them when using the quotient rule.

Rational Functions

A rational function is any function that can be expressed as the ratio of two polynomials \(\frac{P(x)}{Q(x)}\). In other words, these functions are quotients of polynomials, just like fractions are quotients of integers. This trait makes them important in calculus because rules like the quotient rule apply directly.

Rational functions might not be straightforward to differentiate because different parts of the fraction might have different complexities. Often, they involve simplifying after applying differentiation rules such as the quotient rule. This is especially true when the polynomials involve other functions, like trigonometric ones.

In the exercise, the given function is rational \(h(x) = \frac{3}{\tan(2x) - x}\), with the numerator being a constant polynomial and the denominator being a composite of a trigonometric and a linear function. When differentiating such functions:

• The presence of trigonometric functions in \(Q(x)\) demands careful attention to their derivatives.
• Ensuring clarity in the integration of trigonometric derivatives results in simplified solutions.
• The simplification process may require recognizing trigonometric identities or algebraic manipulation.

Having a comprehensive understanding of rational functions allows for insightful manipulation of these quotient forms during differentiation, especially when paired with the quotient rule.