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Chapter 1: Problem 39

Differentiate each function. $$y=\frac{\sqrt[3]{x}-7}{\sqrt{x}+3}$$

Short Answer

Expert verified
The derivative \[ y' = \frac{ \frac{1}{3} x^{-1/6} + x^{-2/3} - \frac{1}{2} x^{-1/6} + \frac{7}{2} x^{-1/2} }{ (x^{1/2} + 3)^2 } \]

Step by step solution

01

Rewrite the function using exponents

Rewrite the given function in a form that makes differentiation easier. The original function is given by \[ y = \frac{\sqrt[3]{x} - 7}{\sqrt{x} + 3} \]Rewrite the radicals using exponents: \[ y = \frac{x^{1/3} - 7}{x^{1/2} + 3} \]
02

Use the quotient rule

The quotient rule states that if you have a function \( \frac{u(x)}{v(x)} \), its derivative is given by: \[ y' = \frac{u'v - uv'}{v^2} \] Identify \(u\) and \(v\): \[ u = x^{1/3} - 7 \] \[ v = x^{1/2} + 3 \]
03

Compute the derivatives of the numerator and denominator

Differentiate the numerator and denominator functions. Differentiate \(u\) with respect to \(x\): \[ u' = \frac{d}{dx} (x^{1/3} - 7) = \frac{1}{3}x^{-2/3} \] Differentiate \(v\) with respect to \(x\): \[ v' = \frac{d}{dx}(x^{1/2} + 3) = \frac{1}{2}x^{-1/2} \]
04

Substitute into the quotient rule

Plug the derivatives \(u'\) and \(v'\), as well as the functions \(u\) and \(v\) into the quotient rule formula: \[ y' = \frac{(\frac{1}{3}x^{-2/3})(x^{1/2} + 3) - (x^{1/3} - 7)(\frac{1}{2}x^{-1/2})}{(x^{1/2} + 3)^2} \]
05

Simplify the expression

Simplify the numerator of the expression: \[ y' = \frac{ \frac{1}{3} x^{-2/3} x^{1/2} + \frac{1}{3} x^{-2/3} \times 3 - \frac{1}{2} x^{1/3} x^{-1/2} + \frac{7}{2} x^{-1/2} }{ (x^{1/2} + 3)^2 } \] Combine and simplify the terms: \[ y' = \frac{ \frac{1}{3} x^{-1/6} + x^{-2/3} - \frac{1}{2} x^{-1/6} + \frac{7}{2} x^{-1/2} }{ (x^{1/2} + 3)^2 } \]

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Key Concepts

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

Derivative
In mathematics, the derivative represents the rate at which a function is changing at any given point. When we talk about differentiation, we are finding this rate of change. This concept is fundamental in calculus.

Derivatives help us understand how a function behaves. For instance, they can tell us about the slope of a curve or how a physical quantity like velocity changes over time.

The derivative of a function is denoted as either \(f'(x)\) or \(\frac{df}{dx}\). Finding the derivative involves applying differentiation rules, such as the power rule, product rule, and chain rule. In our exercise, the quotient rule is used to differentiate a ratio of two functions.
Quotient Rule
The quotient rule is a technique for differentiating functions that are divided by each other. It states that if you have a function \(y = \frac{u(x)}{v(x)}\), the derivative \(y'\) is given by:

\[ y' = \frac{u'v - uv'}{v^2} \]

This formula helps us find the rate of change for ratios of functions. To apply the quotient rule, you must:

  • Identify the numerator \(u(x)\) and the denominator \(v(x)\).
  • Find the derivatives \(u'\) and \(v'\) respect to \(x\).
  • Substitute these derivatives into the quotient rule formula.
  • Simplify the resulting expression.


In the given exercise, we identified \(u(x) = x^{1/3} - 7\) and \(v(x) = x^{1/2} + 3\), then computed their derivatives. This allowed us to apply the quotient rule for the final differentiation.
Exponents
Exponents represent the number of times a base number is multiplied by itself. They are a fundamental concept in mathematics and essential in calculus.

In the given exercise, we encounter exponents such as \(x^{1/3}\) and \(x^{1/2}\). To make differentiation easier, we first rewrite radical expressions using exponents.

Some important properties of exponents include:
  • \(a^{m} \cdot a^{n} = a^{m+n}\)
  • \(\frac{a^{m}}{a^{n}} = a^{m-n}\)
  • \((a^{m})^{n} = a^{m \cdot n}\)
  • \(a^{0} = 1\) for any non-zero \(a\)


By converting radicals to exponents, we can apply the power rule for differentiation more easily. For example, the derivative of \(x^{n}\) is \(nx^{n-1}\). This principle was used to find the derivatives of \(u\) and \(v\) in our exercise.

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Most popular questions from this chapter

First, use the Chain Rule to find the answer. Next, check your answer by finding \(f(g(x))\) taking the derivative, and substituting. \(f(u)=\sqrt[3]{u}, g(x)=u=1+3 x^{2}\) Find \((f \circ g)^{\prime}(2)\)

A total-cost function is given by \(C(x)=2000\left(x^{2}+2\right)^{1 / 3}+700\) where \(C(x)\) is the total cost, in thousands of dollars, of producing \(x\) items. Find the rate at which total cost is changing when 20 items have been produced.

The population \(P\), in thousands, of a small city is given by \(P(t)=\frac{500 t}{2 t^{2}+9}\) where \(t\) is the time, in years. (GRAPH CANNOT COPY) a) Find the growth rate. b) Find the population after 12 yr. c) Find the growth rate at \(t=12\) yr.

The function \(f(x)=x^{3}+a x\) is always increasing if \(a>0,\) but not if \(a<0 .\) Use the derivative of \(f\) to explain why this observation is true.

Use the derivative to help show whether each function is always increasing, always decreasing, or neither. $$f(x)=x^{5}+x^{3}$$
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