Problem 10 - Calculus Early Transcendentals

Question

Find \(f^{\prime}(x)\) $$ f(x)=\cos x-x \csc x $$

Step-by-Step Solution

Answer

\(f'(x) = -\sin x - \csc x + x \csc x \cot x\)

Step 1: Understand the Derivative Rules

To find the derivative of the function \(f(x) = \cos x - x \csc x\), we need to apply the basic rules of differentiation, such as the derivative of trigonometric functions and the product rule.

Step 2: Differentiate \(\cos x)\)

The derivative of \(\cos x\) with respect to \(x\) is \(f'(x) = -\sin x\).

Step 3: Differentiate \(x \csc x\) using the Product Rule

The term \(x \csc x\) is a product of two functions: \(x\) and \(\csc x\). We'll use the product rule which states that \((uv)' = u'v + uv'\) for functions \(u(x)\) and \(v(x)\). Here, \(u = x\) and \(v = \csc x\).

Step 4: Find Derivative of \(x\)

The derivative of \(x\) with respect to \(x\) is \(1\).

Step 5: Find Derivative of \(\csc x\)

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

Step 6: Apply the Product Rule

Using the product rule, \( \frac{d}{dx}(x \csc x) = (1)(\csc x) + (x)(-\csc x \cot x) = \csc x - x \csc x \cot x \).

Step 7: Combine the Derivatives

Combine the results from Step 2 and Step 6. The derivative of the whole function is \(-\sin x - (\csc x - x \csc x \cot x)\).

Step 8: Simplify the Expression

Simplify the expression: \(f'(x) = -\sin x - \csc x + x \csc x \cot x\).

Key Concepts

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. These functions are essential in various areas of mathematics, including calculus, where they help in finding derivatives and integrals.

Some of the basic trigonometric functions include:

• Sine (\( \sin x \))
• Cosine (\( \cos x \))
• Tangent (\( \tan x \))
• Cosecant (\( \csc x \)): \( \csc x = \frac{1}{\sin x} \)
• Secant (\( \sec x \)): \( \sec x = \frac{1}{\cos x} \)
• Cotangent (\( \cot x \)): \( \cot x = \frac{1}{\tan x} \)

The derivatives of these functions have specific formulas. For example, the derivative of \( \cos x \) is \(-\sin x\). Similarly, the derivative of \( \csc x \) is \(-\csc x\cot x\). Understanding these derivatives is crucial as they are often used in calculus problems involving trigonometric functions.

Product Rule

The product rule is a differentiation rule used when taking the derivative of a product of two functions. This rule is essential when you have an expression that is the product of two separate parts, each of which depends on the variable you are differentiating.

The product rule can be stated as:\[ (uv)' = u'v + uv' \]Here, \(u(x)\) and \(v(x)\) represent the functions involved in a product. Its application involves three steps:

• Find the derivative of the first function \(u(x)\)
• Find the derivative of the second function \(v(x)\)
• Apply the formula by multiplying derivatives and summing accordingly

For example, in the term \(x \csc x\), \(u = x\) and \(v = \csc x\). The derivatives would be \(u' = 1\) and \(v' = -\csc x \cot x\). Applying the product rule gives:\[ \frac{d}{dx}(x \csc x) = 1 \cdot \csc x + x \cdot (-\csc x \cot x)\]Understanding and applying the product rule is essential in solving problems involving derivatives of products of functions in calculus.

Differentiation Rules

Differentiation involves finding the derivative of a function, a fundamental idea in calculus used to determine the rate at which one quantity changes with respect to another. Different types of functions and their interactions require different rules to find derivatives effectively.

Important differentiation rules include:

• Power Rule: For \(f(x) = x^n\), the derivative \(f'(x) = n x^{n-1}\).
• Constant Multiple Rule: If \(f(x) = c g(x)\), then \(f'(x) = c g'(x)\).
• Sum and Difference Rule: The derivative of a sum/difference is the sum/difference of the derivatives.
• Chain Rule: Used for composite functions, helping to differentiate nested functions.
• Product Rule: Essential for differentiating products of functions, as seen with \(x \csc x\).
• Quotient Rule: For functions as ratios, useful when dealing with trigonometric ratios.
• Trigonometric Derivatives: Include rules for \(\sin x\), \(\cos x\), \(\tan x\), and their reciprocals.

By understanding and applying these rules, finding derivatives of complex expressions becomes more manageable. Mastery of these foundational tools is crucial for success in higher-level calculus problems.