/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 20 find the third derivative of the... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 20

find the third derivative of the function. $$ f(x)=\left(x^{3}-6\right)^{4} $$

Short Answer

Expert verified
The third derivative of the function \( f(x) = (x^{3}-6)^{4} \) is \( f'''(x) = 36(x^{3}-6)^{2} \cdot (30x^{4} + 28x^{2} - 3) \).

Step by step solution

01

Find the First Derivative

Differentiate \( f(x) = (x^{3}-6)^{4} \) using the chain rule. The chain rule, in this case, would be applied as follows: the derivative of the outside function, keeping the inside function the same, multiplied by the derivative of the inside function. This gives \( f'(x) = 4(x^{3}-6)^{3} \cdot 3x^{2} \).
02

Find the Second Derivative

For \( f''(x) \), differentiate the first derivative \( f'(x) = 12x^{2}(x^{3}-6)^{3} \) using the chain rule and product rule. Here, let \( u=x^{3}-6 \) and \( v=x^{2} \). Then \( f''(x) \) becomes \( f''(x) = 36x(x^{3}-6)^{2}\cdot(x^{2}+4x^{4}) \).
03

Find the Third Derivative

Finally, for the third derivative, \( f'''(x) \), differentiate the second derivative \( f''(x) = 36x(x^{3}-6)^{2}\cdot(x^{2}+4x^{4}) \) using the chain rule and product rule once again. This gives \( f'''(x) = 36(x^{3}-6)^{2} \cdot (30x^{4} + 28x^{2} - 3) \).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Chain Rule
The chain rule is an essential tool for differentiating composite functions. Composite functions are functions nested within each other, like in the function given: \( f(x) = (x^3 - 6)^4 \). If we consider this as a function \(g\) inside another function \(h\), then \(g(x) = x^3 - 6\) and \(h(u) = u^4\), where \(u = g(x)\).
The chain rule states that the derivative of \(h(g(x))\) is the derivative of \(h(u)\) with respect to \(u\), times the derivative of \(g(x)\) with respect to \(x\). This looks like:
  • Derivative of the outer function: \(h'(u) = 4u^3\)
  • Derivative of the inner function: \(g'(x) = 3x^2\)
  • Apply the chain rule: \(f'(x) = 4(x^3 - 6)^3 \, (3x^2)\)
The chain rule helps simplify the process of handling and differentiating these complicated functions with nested layers.
Product Rule
The product rule is employed when differentiating functions that are products of two separate functions. For example, when finding the second derivative in the given problem, we encountered \(f'(x) = 12x^2(x^3 - 6)^3\).
In this scenario, think of these as two functions, \(u\) and \(v\), where \(u(x) = 12x^2\) and \(v(x) = (x^3 - 6)^3\). The product rule states that to differentiate \(u(x)v(x)\), you do:
  • \( (uv)' = u'v + uv' \)
  • \( u'(x) = 24x \)
  • \( v'(x) = 3(x^3 - 6)^2 \, (3x^2) \)
Using the product rule, this simplifies to: \[ f''(x) = 24x(x^3 - 6)^3 + 12x^2 \cdot 3(x^3 - 6)^2 \cdot 3x^2 \]The product rule ensures each part of the product is considered and accurately differentiated, maintaining the integrity of the function.
Higher-Order Derivatives
Higher-order derivatives involve finding the second, third, or even higher derivatives of a function. These derivatives reveal information about the behavior and curvature of the function at various points.
In the given exercise, the third derivative \(f'''(x)\) is found by differentiating the second derivative \(f''(x) = 36x(x^3 - 6)^2 \cdot (x^2 + 4x^4)\). In this step:
  • Carefully apply both the product and chain rules again.
  • Ensure each differentiation process is followed correctly.
  • The result gives the third derivative: \( f'''(x) = 36(x^3-6)^2 \cdot (30x^4 + 28x^2 - 3) \)
Having the third derivative helps understand how the rate of change of the rate of change (acceleration, for instance) behaves. It鈥檚 particularly valuable in physics and dynamics studies where such details are crucial.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find the derivative of the function. $$ f(t)=-3 t^{2}+2 t-4 $$

(a) sketch the graphs of \(f\) and \(g,(b)\) find \(f^{\prime}(1)\) and \(g^{\prime}(1),(c)\) sketch the tangent line to each graph when \(x=1,\) and \((d)\) explain the relationship between \(f^{\prime}\) and \(g^{\prime}\). $$ \begin{array}{l}{f(x)=x^{3}} \\ {g(x)=x^{3}+3}\end{array} $$

Find the marginal profit for producing units. (The profit is measured in dollars.) $$ P=-0.00025 x^{2}+12.2 x-25,000 $$

Use a graphing utility to graph \(f\) and \(f^{\prime}\) over the given interval. Determine any points at which the graph of \(f\) has horizontal tangents. $$f(x)=4.1 x^{3}-12 x^{2}+2.5 x\quad [0,3] $$

Find the derivative of the function. $$ f(x)=-2 $$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.