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Chapter 3: Problem 17

Differentiate the functions. $$y=\frac{1}{\pi}+\frac{2}{x^{2}+1}$$

Short Answer

Expert verified
The derivative of \( y = \frac{1}{\pi} + \frac{2}{x^2 + 1} \) is \( y' = \frac{-4x}{(x^2 + 1)^2} \).

Step by step solution

01

Identify the terms in the function

The function given is \(y = \frac{1}{\begin{array}{l}\pi}\end{array} + \frac{2}{x^{2}+1}\). There are two terms to differentiate: the constant term \( \frac{1}{\begin{array}{l}\pi\end{array}} \) and the rational function \( \frac{2}{x^{2}+1} \).
02

Differentiate the constant term

The differentiation of a constant is always 0. Therefore, \( \frac{d}{dx}\left( \frac{1}{\begin{array}{l}\pi \end{array}}\right) = 0 \).
03

Differentiate the rational function

Apply the differentiation rules to the second term. For the term \( \frac{2}{x^{2}+1} \), we use the formula \( \frac{d}{dx} \, \frac{u}{v} = \frac{v \cdot u' - u \cdot v'}{v^2} \). Here, \( u = 2 \) and \( v = x^2 + 1 \). First compute the derivatives of \( u \) and \( v \). Since \( u \) is a constant, \( u' = 0 \). The derivative of \( v \), \( v' \), is \( 2x \). Therefore, \( \frac{d}{dx} \, \frac{2}{x^{2}+1} = \frac{(x^2 + 1) \cdot 0 - 2 \cdot 2x}{(x^2 + 1)^2} = \frac{-4x}{(x^2 + 1)^2}\).
04

Combine the results

Add up the results of each term's differentiation: \( y' = 0 + \frac{-4x}{(x^2 + 1)^2} \). Simplifying this gives us \( y' = \frac{-4x}{(x^2 + 1)^2} \).

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Key Concepts

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

constant term differentiation
When differentiating a constant term, the result is always zero. This is because a constant term does not change regardless of the value of the variable it is associated with. In our example, we have the constant term \( \frac{1}{\pi} \). Since \( \pi \) is a constant, the derivative of \( \frac{1}{\pi} \) is \( 0 \. \)
  • Any term that does not include the variable being differentiated will have a derivative of zero.
  • This simplification greatly reduces the complexity of the differentiation process.
So, \( \frac{d}{dx} \left( \frac{1}{\pi} \right) = 0 \).
rational function differentiation
Rational function differentiation involves the process of finding the derivative of a function that is expressed as the ratio of two polynomials. In our problem, the rational function is \( \frac{2}{x^2 + 1} \). To differentiate such functions, we often use the quotient rule.
  • Identify the numerator and denominator of the rational function. Here, \( u = 2 \) and \( v = x^2 + 1 \).
  • Compute the derivatives of both parts: \( u' = 0 \) (since \( u \) is a constant) and \( v' = 2x \).
  • Apply the quotient rule: \( \frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v \cdot u' - u \cdot v'}{v^2} \).
This becomes \( \frac{(x^2 + 1) \cdot 0 - 2 \cdot 2x}{(x^2 + 1)^2} = \frac{-4x}{(x^2 + 1)^2} \).
power rule
The power rule is a basic but powerful rule in differentiation. It states that for any function \( f(x) = x^n \), the derivative \( f'(x) \) is \( nx^{n-1} \. \) This rule applies to any real number \( n \. \) While our exercise doesn't directly use the power rule on typical polynomial expressions, understanding it helps in broader contexts where variables are raised to powers.
  • For example, if \( f(x) = x^3 \, \) then \( f'(x) = 3x^2 \).
  • Whenever you see such terms, apply the power rule to quickly find the derivative.
The power rule is efficient and simplifies differentiation tasks significantly.
quotient rule in differentiation
The quotient rule is used when differentiating a function that is the ratio of two functions. Given that \( y = \frac{u}{v} \, \) the quotient rule states: \( \frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v \cdot u' - u \cdot v'}{v^2} \. \) This method was directly applied in our problem.
  • First, identify \( u \) and \( v \) in your function. Here, \( u = 2 \) and \( v = x^2 + 1 \).
  • Next, find the derivatives \( u' \) and \( v' \). \ Since \( u \) is a constant, \ u' = 0. \ The derivative of \( v \) is \( v' = 2x \).
  • Substitute these values into the quotient rule formula: \( \frac{d}{dx} \frac{2}{x^2 + 1} = \frac{(x^2 + 1) \cdot 0 - 2 \cdot 2x}{(x^2 + 1)^2} = \frac{-4x}{(x^2 + 1)^2} \).
The quotient rule simplifies the process of differentiating complex rational functions.

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Most popular questions from this chapter

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