/*! 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 70 Find the following higher-order ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 70

Find the following higher-order derivatives. $$\left.\frac{d^{3}}{d x^{3}}\left(x^{4.2}\right)\right|_{x=1}$$

Short Answer

Expert verified
Answer: The third derivative of the function evaluated at x = 1 is $$29.568$$.

Step by step solution

01

Find the first derivative of the function

To find the first derivative of the function, we can make use of the power rule which states that: $$\frac{d(f(x))}{dx} = nx^{n-1}$$ Applying the power rule, the first derivative, f'(x), will be: $$f'(x) = 4.2x^{4.2-1}$$ Simplify: $$f'(x) = 4.2x^{3.2}$$
02

Find the second derivative of the function

Now that we have the first derivative, we can use the same power rule to find the second derivative, f''(x): $$f''(x) = 3.2\cdot 4.2x^{3.2-1}$$ Simplify: $$f''(x) = 13.44x^{2.2}$$
03

Find the third derivative of the function

To find the third derivative, f'''(x), we apply the power rule one more time: $$f'''(x) = 2.2\cdot 13.44x^{2.2-1}$$ Simplify: $$f'''(x) = 29.568x^{1.2}$$
04

Evaluate the third derivative at x = 1

Now that we have the third derivative, we will evaluate it at x = 1: $$\left.\frac{d^{3}}{d x^{3}}\left(x^{4.2}\right)\right|_{x=1} = 29.568(1)^{1.2}$$ Since 1 raised to any power remains 1, we have: $$\left.\frac{d^{3}}{d x^{3}}\left(x^{4.2}\right)\right|_{x=1} = 29.568$$ Thus, the solution for the problem is $$29.568$$.

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!

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

Special Product Rule In general, the derivative of a product is not the product of the derivatives. Find nonconstant functions \(f\) and \(g\) such that the derivative of \(f g\) equals \(f^{\prime} g^{\prime}\).

Consider the following functions (on the given interval, if specified). Find the inverse function, express it as a function of \(x,\) and find the derivative of the inverse function. $$f(x)=x^{2 / 3}, \text { for } x>0$$

Use any method to evaluate the derivative of the following functions. $$h(r)=\frac{2-r-\sqrt{r}}{r+1}$$

One of the Leibniz Rules One of several Leibniz Rules in calculus deals with higher-order derivatives of products. Let \((f g)^{(n)}\) denote the \(n\) th derivative of the product \(f g,\) for \(n \geq 1\) a. Prove that \((f g)^{(2)}=f^{\prime \prime} g+2 f^{\prime} g^{\prime}+f g^{\prime \prime}\) b. Prove that, in general,$$(f g)^{(n)}=\sum_{k=0}^{n}\left(\begin{array}{l} n \\\k\end{array}\right) f^{(k)} g^{(n-k)}$$ where \(\left(\begin{array}{l}n \\\ k\end{array}\right)=\frac{n !}{k !(n-k) !}\) are the binomial coefficients. c. Compare the result of (b) to the expansion of \((a+b)^{n}\).

Orthogonal trajectories Two curves are orthogonal to each other if their tangent lines are perpendicular at each point of intersection (recall that two lines are perpendicular to each other if their slopes are negative reciprocals. . A family of curves forms orthogonal trajectories with another family of curves if each curve in one family is orthogonal to each curve in the other family. For example, the parabolas \(y=c x^{2}\) form orthogonal trajectories with the family of ellipses \(x^{2}+2 y^{2}=k,\) where \(c\) and \(k\) are constants (see figure). Use implicit differentiation if needed to find \(d y / d x\) for each equation of the following pairs. Use the derivatives to explain why the families of curves form orthogonal trajectories. \(y=m x ; x^{2}+y^{2}=a^{2},\) where \(m\) and \(a\) are constants
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

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.