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Chapter 8: Problem 10

In Exercises, find the second derivative of the function. $$ f(x)=x \sqrt[3]{x} $$

Short Answer

Expert verified
The second derivative of the function \( f(x) = x \sqrt[3]{x} \) is equal to \( f''(x) = \frac{4}{9}*x^{-\frac{2}{3}} \).

Step by step solution

01

Express Function

Our first step is to rewrite the function in the form \( f(x) = x^\frac{4}{3} \).
02

First Derivative

Next, we find the first derivative using the power rule (where \( f'(x) = n*x^{n-1} \)) which results in \( f'(x) = \frac{4}{3}*x^{\frac{4}{3} - 1} = \frac{4}{3}*x^{\frac{1}{3}} \).
03

Second Derivative

Now, we differentiate the first derivative to find the second derivative, again using the power rule. This gives us \( f''(x) = \frac{4}{3}*\frac{1}{3}*x^{\frac{1}{3}-1} = \frac{4}{9}*x^{-\frac{2}{3}} \).

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Key Concepts

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

Calculus
Calculus is a branch of mathematics that studies changes. It is widely used in various fields like physics, engineering, economics, and more. Calculus consists of two main components: differentiation and integration. Differentiation helps us understand how things change, while integration is about the accumulation of quantities. In this problem, we are focusing on differentiation.
  • Differentiation: This process allows us to find the rate at which a function is changing at any given point.
  • Integration: This is the reverse process of differentiation. It involves finding the original function given its rate of change.
Understanding calculus helps us predict and analyze the behavior of complex systems, and enables us to solve diverse problems ranging from simple motions to intricate dynamic systems.
Power Rule
The power rule is crucial in differentiation, especially when working with polynomial functions. It provides a quick way to find derivatives of functions of the form \(x^n\), where \(n\) is a real number.
To apply the power rule:
  • Identify the exponent \(n\) in the term \(x^n\).
  • Multiply the term by the exponent to get \(n*x^{n-1}\).
  • Subtract one from the exponent \(n\).
For the function \(f(x) = x^{\frac{4}{3}}\), we used the power rule to determine that \(f'(x) = \frac{4}{3}*x^{\frac{1}{3}}\). Applying the power rule again to this result gives us the second derivative, \(f''(x) = \frac{4}{9}*x^{-\frac{2}{3}}\). This rule simplifies the differentiation process and is fundamental in calculus.
Function Differentiation
Function differentiation is the process of finding a new function, called the derivative, which tells us the rate of change of the original function. Differentiating helps us determine how a function behaves locally by providing insights into its slope or gradient.
To differentiate a function:
  • Express the function in a form that's easily differentiable, often using algebraic manipulation.
  • Use differentiation techniques, such as the power rule, to find the derivative.
  • Apply the procedure repeatedly for higher-order derivatives, like the second derivative.
In the exercise, we first expressed \(f(x) = x\sqrt[3]{x}\) as \(f(x) = x^{\frac{4}{3}}\) to make it easier to handle. Differentiating once gave us \(f'(x)\), and differentiating again yielded \(f''(x)\), which shows how the rate of change itself changes. Such analysis is essential in understanding more about the function's behavior and anticipating its response to different inputs.

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Most popular questions from this chapter

The resident population \(P\) (in millions) of the United States from 1790 through 2000 can be modeled by \(P=0.00000583 t^{3}+0.005003 t^{2}+0.13776 t+4.658\) \(-10 \leq t \leq 200\), where \(t=0\) corresponds to 1800 (a) Make a conjecture about the maximum and minimum populations in the U.S. from 1790 to 2000 . (b) Analytically find the maximum and minimum populations over the interval. (c) Write a brief paragraph comparing your conjecture with your results in part (b).

In Exercises, find the point(s) of inflection of the graph of the function. $$ f(x)=(x-1)^{3}(x-5) $$

In Exercises, find the absolute extrema of the function on the closed interval. Use a graphing utility to verify your results. $$ g(x)=4\left(1+\frac{1}{x}+\frac{1}{x^{2}}\right), \quad[-4,5] $$

In Exercises, sketch a graph of a function \(f\) having the given characteristics. \begin{aligned} &f(2)=f(4)=0 \\ &f^{\prime}(x)<0 \text { if } x<3 \\ &f^{\prime}(3)=0 \\ &f^{\prime}(x)>0 \text { if } x>3 \\ &f^{\prime}(x)>0 \end{aligned}

In Exercises, find the absolute extrema of the function on the closed interval. Use a graphing utility to verify your results. $$ f(x)=x^{2}+2 x-4, \quad[-1,1] $$
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