/*! 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 72 Find the following higher-order ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 72

Find the following higher-order derivatives. $$\frac{d^{n}}{d x^{n}}\left(2^{x}\right)$$

Short Answer

Expert verified
Answer: The general formula for the $$n^{th}$$ order derivative of the function $$2^x$$ with respect to $$x$$ is $$\frac{d^n}{dx^n}\left(2^x\right) = 2^x (\ln2)^n$$.

Step by step solution

01

First derivative

To find the first derivative, we start by differentiating the function $$2^x$$ with respect to $$x$$: $$\frac{d}{dx}\left(2^x\right) = 2^x \ln2$$
02

Second derivative

Now we find the second derivative, by differentiating the first derivative with respect to $$x$$: $$\frac{d^2}{dx^2}\left(2^x\right) = \frac{d}{dx}\left(2^x \ln2\right) = 2^x (\ln2)^2$$
03

Third derivative

Let's find the third derivative by differentiating the second derivative: $$\frac{d^3}{dx^3}\left(2^x\right) = \frac{d}{dx}\left(2^x (\ln2)^2\right) = 2^x (\ln2)^3$$
04

Identify the pattern

At this point, we can observe a pattern in the derivatives: 1. The first derivative is $$2^x \ln2$$ 2. The second derivative is $$2^x (\ln2)^2$$ 3. The third derivative is $$2^x (\ln2)^3$$ We can generalize the pattern for any order derivative:
05

Generalize for the $$n^{th}$$ derivative

Following the observed pattern, the $$n^{th}$$ derivative of the function $$2^x$$ can be written as: $$\frac{d^n}{dx^n}\left(2^x\right) = 2^x (\ln2)^n$$

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

Product Rule for the second derivative Assuming the first and second derivatives of \(f\) and \(g\) exist at \(x\), find a formula for \(\frac{d^{2}}{d x^{2}}(f(x) g(x))\)

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. $$f(x)=\sqrt{x+2}, \text { for } x \geq-2$$

The following limits equal the derivative of a function \(f\) at a point a. a. Find one possible \(f\) and \(a\) b. Evaluate the limit. $$\lim _{x \rightarrow \pi / 4} \frac{\cot x-1}{x-\frac{\pi}{4}}$$

Prove the following identities and give the values of \(x\) for which they are true. $$\sin \left(2 \sin ^{-1} x\right)=2 x \sqrt{1-x^{2}}$$

Visualizing tangent and normal lines a. Determine an equation of the tangent line and normal line at the given point \(\left(x_{0}, y_{0}\right)\) on the following curves. (See instructions for Exercises \(63-68 .)\) b. Graph the tangent and normal lines on the given graph. $$\begin{aligned}&3 x^{3}+7 y^{3}=10 y\\\&\left(x_{0}, y_{0}\right)=(1,1)\end{aligned}$$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

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.