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Chapter 8: Problem 21

Find the following: $$ \frac{d}{d x}\left(\frac{\pi}{\pi x+\pi}\right) $$

Short Answer

Expert verified
The derivative of the given function with respect to \(x\) is \(- \pi / (\pi x+\pi)\).

Step by step solution

01

Identify \(u\) and \(v\)

We first identify \(u\) and \(v\) in our expression. In our case, \(u=\pi\) and \(v=\pi x+\pi\). We then also calculate \(du/dx\) and \(dv/dx\). For any constant \(c\), \(dc/dx = 0\), so \(du/dx=0\). For \(v\), \(dv/dx = \pi\) as the derivative of \(x\) is \(1\) and of \(\pi\) is \(0\).
02

Apply the quotient rule

We then use the quotient rule to get: \(d/dx(u/v) = (v*(du/dx) - u * (dv/dx)) / (v^2)\). Plugging in our \(u\), \(v\), \(du/dx\), and \(dv/dx\), we get: \((\pi x+\pi)*(0) - \pi*\pi / (\pi x+\pi)^2 = - \pi^2 / (\pi x+\pi)^2\).
03

Simplify the expression

The expression simplifies to \(- \pi / (\pi x+\pi)\). In this step we took advantage of the fact that \(\pi \neq 0\) so we could divide both the numerator and denominator by \(\pi\) to simplify the expression.

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Key Concepts

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

Quotient Rule
The quotient rule is an essential tool in calculus when you want to differentiate a function that is the ratio of two differentiable functions. It's particularly useful because not every function we encounter is a simple product or sum.

The general form of the quotient rule is given by: \[\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v\left(\frac{du}{dx}\right) - u\left(\frac{dv}{dx}\right)}{v^2}\] where \(u\) and \(v\) are functions of \(x\), and \(v eq 0\) to avoid division by zero.
  • Identify \(u\) as the numerator of the function and \(v\) as the denominator.
  • Find the derivatives \(du/dx\) and \(dv/dx\) of these functions.
  • Substitute these into the quotient rule formula.

    • This approach allows you to handle more complex expressions by breaking them down into simpler parts. Practicing with the quotient rule can enhance your skills in tackling challenging differentiation problems in your calculus coursework.
Differentiation
Differentiation is a key concept in calculus that involves finding the derivative of a function. The derivative represents the rate of change or the slope of the function at any given point.

Let's dive into why understanding differentiation is so important:
  • The derivative of a constant is zero. For example, if \(u\) is a constant like \(\pi\), then \(du/dx = 0\).
  • Differentiation of powers means you can apply standard rules like the power rule to determine the derivative of variable terms.
  • Understanding how to differentiate simple functions makes it easier to apply rules like the quotient rule effectively.

Differentiation provides us a way to model and predict behavior in dynamic systems, such as velocity, growth rates, and other real-world applications where understanding change is crucial.
Calculus Steps
When solving calculus problems, following a structured approach can help you navigate through the solution effectively. This is especially important in differentiation. Let's reflect on the typical steps involved in solving a quotient differentiation problem:
  • First, identify the functions forming the numerator \(u\) and denominator \(v\) in your problem.
  • Next, calculate the derivatives \(du/dx\) and \(dv/dx\) of these components separately.
  • Apply the quotient rule using the derivatives and original functions you've identified.
  • Finally, simplify the expression by reducing terms and combining like parts if possible, to reach the cleanest form of the derivative.

This structured approach helps ensure that you do not overlook any part of the process, making it easier to find mistakes and correct them efficiently. Practice each step diligently to build your confidence and proficiency in calculus.

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Most popular questions from this chapter

The function \(f(x)=x+|x|\) is continuous at \(x=0\) but not differentiable at \(x=0\). Explain, using the de nitions of continuity at a point and differentiability at a point.

Find \(f^{\prime}(x) .\) Strategize to minimize your work. For example, \(\frac{x^{2}+3}{3 x}\) does not require the Quotient Rule. \(\frac{x^{2}+3}{3 x}=\frac{x}{3}+\frac{1}{x}=\frac{1}{3} x+x^{-1} .\) This is simpler to differentiate. $$ f(x)=\left(\frac{5 x^{2}}{2}+7 x^{5}-5 x\right) x $$

Find the following: $$ \frac{d}{d x}\left(\frac{x^{2}+5 x}{2 x^{10}}\right) $$

Find the equation of the tangent line to \(f(x)=x\left(x^{2}+2\right)\) at \(x=1\).

Find \(f^{\prime}(x) .\) Strategize to minimize your work. For example, \(\frac{x^{2}+3}{3 x}\) does not require the Quotient Rule. \(\frac{x^{2}+3}{3 x}=\frac{x}{3}+\frac{1}{x}=\frac{1}{3} x+x^{-1} .\) This is simpler to differentiate. $$ f(x)=\frac{x+2}{x} $$
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