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Chapter 1: Problem 13

Find \(f^{\prime \prime}(x)\). $$f(x)=x^{4}+\frac{3}{x}$$

Short Answer

Expert verified
f''(x) = 12x^2 + 6x^{-3}

Step by step solution

01

Differentiate the function

To find the second derivative, first find the first derivative of the function. The function is given as \[ f(x) = x^4 + \frac{3}{x} \]. Differentiate each term separately. The derivative of \( x^4 \) is \( 4x^3 \). Rewrite \( \frac{3}{x} \) as \( 3x^{-1} \) and differentiate it using the power rule: \( \frac{d}{dx}[3x^{-1}] = -3x^{-2} \). So, \[ f'(x) = 4x^3 - 3x^{-2} \].
02

Differentiate the first derivative

To find the second derivative, differentiate the first derivative \( f'(x) = 4x^3 - 3x^{-2} \). Differentiate each term separately. The derivative of \( 4x^3 \) is \( 12x^2 \). The derivative of \( -3x^{-2} \) is \( 6x^{-3} \) using the power rule. So, \[ f''(x) = 12x^2 + 6x^{-3} \].

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Key Concepts

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

Differentiation
Differentiation is a fundamental concept in calculus. It deals with finding the derivative of a function, which is essentially a measure of how a function changes as its input changes. The derivative provides the rate at which one quantity changes with respect to another. For example, in the context of our exercise, we differentiate the function \( f(x) = x^4 + \frac{3}{x} \). This process of differentiation helps us find the first derivative, \( f'(x) \), and subsequently the second derivative, \( f''(x) \). The method involves applying differentiation rules like the power rule to each term in the function.

Differentiation has multiple applications:
  • Determining the slope of a function at any point
  • Finding critical points to understand maximum and minimum values
  • Analyzing the behavior of functions for optimization problems
Power Rule
The power rule is a simple and essential rule in differentiation. It states that for any function in the form \( f(x) = x^n \), the derivative is \( f'(x) = nx^{n-1} \). This rule makes it straightforward to differentiate polynomial functions.

In our exercise, we use the power rule multiple times:
  • First, to differentiate \( x^4 \), resulting in \( 4x^3 \)
  • Then, to differentiate \( 3x^{-1} \), resulting in \( -3x^{-2} \)
  • Finally, in the second derivative step, where we differentiate each term in the first derivative
Using the power rule, we derive \( f'(x) = 4x^3 - 3x^{-2} \) and \( f''(x) = 12x^2 + 6x^{-3} \). This rule simplifies differentiation greatly, especially for polynomial expressions.
Calculus
Calculus is a branch of mathematics that studies continuous change. It is divided into two main areas: differential calculus and integral calculus. Our exercise focuses on differential calculus, specifically on finding second derivatives.

Some key aspects of calculus include:
  • Understanding derivatives, which represent the rate of change
  • Using integration, which involves finding the area under curves
  • Applying fundamental theorems that link differentiation and integration
Differential calculus is particularly useful for:
  • Analyzing motion and change in various fields like physics and engineering
  • Solving optimization problems in economics and other disciplines
  • Modeling real-world phenomena accurately
In our problem, calculus helps us find the second derivative \( f''(x) \) to better understand the concavity and behavior of the function \( f(x) \).

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Most popular questions from this chapter

Differentiate each function. $$f(x)=\frac{3 x^{2}+2 x}{x^{2}+1}$$

For each of the following, graph \(f\) and \(f^{\prime}\) and then determine \(f^{\prime}(1) .\) For Exercises use Deriv on the \(T I-83\). $$f(x)=x^{4}-x^{3}$$

Differentiate. $$y=\sqrt{(2 x-3)^{2}+1}$$

Tongue-Tied Sauces, Inc., finds that the revenue, in dollars, from the sale of \(x\) bottles of barbecue sauce is given by \(R(x)=7.5 x^{0.7} .\) Find the rate at which average revenue is changing when 81 bottles of barbecue sauce have been produced.

If \(f(x)\) is a function, then \((f \circ f)(x)=f(f(x))\) is the composition of \(f\) with itself. This is called an iterated function, and the composition can be repeated many times. For example, \((f \circ f \circ f)(x)=f(f(f(x))) .\) Iterated functions are very useful in many areas, including finance (compound interest is \(a\) simple case) and the sciences (in weather forecasting, for example). For the each function, use the Chain Rule to find the derivative.. If \(f(x)=x^{2}+1,\) find \(\frac{d}{d x}[(f \circ f \circ f)(x)]\).
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