Problem 31 Compute the derivative of the gi... [FREE SOLUTION]

Chapter 2: Problem 31

Compute the derivative of the given function. $$f(x)=\frac{\cos x}{x}+\frac{x}{\tan x}$$

Short Answer

Expert verified

The derivative is $ f'(x) = \frac{-x \sin x - \cos x}{x^2} + \frac{\tan x - x \sec^2 x}{\tan^2 x} $.

Step by step solution

Identify the Derivative Rule for Each Term

The function is composed of two terms: $ \frac{\cos x}{x} $ and $ \frac{x}{\tan x} $. For each term, we need to apply the quotient rule for derivatives, which is given by: $ \left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2} $.

Derivative of First Term $ \frac{\cos x}{x} $

Let $ u = \cos x $ and $ v = x $. The derivatives are $ u' = -\sin x $ and $ v' = 1 $. Apply the quotient rule:\[ \left( \frac{\cos x}{x} \right)' = \frac{(-\sin x)x - \cos x \cdot 1}{x^2} = \frac{-x \sin x - \cos x}{x^2} \].

Derivative of Second Term $ \frac{x}{\tan x} $

Let $ u = x $ and $ v = \tan x $. The derivatives are $ u' = 1 $ and $ v' = \sec^2 x $. Apply the quotient rule:\[ \left( \frac{x}{\tan x} \right)' = \frac{1 \cdot \tan x - x \cdot \sec^2 x}{\tan^2 x} = \frac{\tan x - x \sec^2 x}{\tan^2 x} \].

Combine Derivatives of Both Terms

The derivative of the function $ f(x) = \frac{\cos x}{x} + \frac{x}{\tan x} $ is the sum of the derivatives of the two terms:\[ f'(x) = \frac{-x \sin x - \cos x}{x^2} + \frac{\tan x - x \sec^2 x}{\tan^2 x} \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quotient Rule

In calculus, the quotient rule is a fundamental method for finding the derivative of a function that is the ratio of two other functions, say $ u(x) $ and $ v(x) $. The quotient rule states that the derivative of $ \frac{u}{v} $ is given by:

$ \left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2} $

This rule is particularly useful when dealing with complex rational expressions where both the numerator and the denominator are functions that vary with respect to $ x $.
To apply the quotient rule, follow these steps:

Identify $ u $ and $ v $ from the function $ \frac{u}{v} $.
Find the derivatives $ u' $ and $ v' $.
Substitute into the quotient rule formula to find $ \left( \frac{u}{v} \right)' $.

The quotient rule can sometimes lead to complex expressions, but with careful algebraic manipulation and simplification, you can usually express the derivative in a more workable form.

Trigonometric Functions

Trigonometric functions such as $ \cos x $, $ \sin x $, $ \tan x $, and $ \sec x $ often appear in calculus problems, especially when dealing with derivatives and integrals.
Understanding their derivatives is crucial. Here are some key derivatives of trigonometric functions:

The derivative of $ \sin x $ is $ \cos x $.
The derivative of $ \cos x $ is $ -\sin x $.
The derivative of $ \tan x $ is $ \sec^2 x $.

Why is this important? Trigonometric derivatives help us analyze oscillations, wave patterns, and other periodic phenomena. They are essential in everything from engineering to physics.
When solving derivatives involving trigonometric functions, keep these in mind:

Ensure you're familiar with basic trigonometric identities, as they can often simplify your work.
These functions have periodic properties, so their behavior can be quite cyclical.
In composite functions, apply the chain rule if required to accurately determine derivatives.

Calculus Problem Solving

Solving calculus problems like finding the derivative of a complex function involves a structured approach. Start by identifying all components of the function being considered. In our exercise, the expression is composed of two quotient forms, $ \frac{\cos x}{x} $ and $ \frac{x}{\tan x} $. Here鈥檚 a simple process to follow for problem-solving in calculus:

Break the problem down: Understand which rules apply to each part of the expression.
Apply relevant rules like the quotient rule, as demonstrated in this exercise.
Carefully calculate each derivative, remembering to substitute back any original variables at the end.
Check your work: Ensure all terms in the derivation have been considered and correctly simplified.
Combine the results: In problems where multiple derivatives are calculated, such as this one, sum the terms as needed.

Proficiency in calculus problem-solving comes with practice and understanding of fundamental concepts. Try to visualize the function and its behavior as you find its derivative. This will give you deeper insight and enable you to anticipate potential pitfalls or simplifications.

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91影视

Compute the derivative of the given function. $$f(x)=\frac{\cos x}{x}+\frac{x}{\tan x}$$

Short Answer

Step by step solution

Identify the Derivative Rule for Each Term

Derivative of First Term \( \frac{\cos x}{x} \)

Derivative of Second Term \( \frac{x}{\tan x} \)

Combine Derivatives of Both Terms

Key Concepts

Quotient Rule

Trigonometric Functions

Calculus Problem Solving

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