/*! 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 6 Evaluate \(\frac{d}{d x}\left(3^... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 6

Evaluate \(\frac{d}{d x}\left(3^{x}\right)\).

Short Answer

Expert verified
Answer: The derivative of the function \(3^x\) with respect to \(x\) is \(3^x \ln{3}\).

Step by step solution

01

Rewrite Function Using Natural Exponential Function and Logarithm Properties

Rewrite the given function \(3^x\) using the natural exponential function (\(e^x\)) and logarithmic properties, which will simplify the derivative. $$ 3^x = e^{x \ln{3}} $$ The property we used here is \(a^x = e^{x \ln{a}}\), where \(a = 3\) in this case.
02

Find Derivative of Exponential Function

Differentiate the rewritten function with respect to \(x\). The derivative of \(e^{x \ln{3}}\) can be found using the chain rule: $$ \frac{d}{dx}\left(e^{x \ln{3}}\right) = e^{x \ln{3}} \frac{d}{dx}(x \ln{3}) $$ Note that we are multiplying by the derivative of the exponent (chain rule).
03

Differentiate the Exponent Function with respect to x

Now, differentiate the exponent, which is \(x \ln{3}\), with respect to \(x\). Using the constant rule and keeping in mind that \(\ln{3}\) is a constant: $$ \frac{d}{dx}(x \ln{3}) = 1 * \ln{3} = \ln{3} $$
04

Substitute the Derivative of the Exponent Back Into the Expression

Plug the derivative of the exponent from Step 3 back into the expression found in Step 2: $$ e^{x \ln{3}} \frac{d}{dx}(x\ln{3})= e^{x \ln{3} } \ln{3} $$
05

Rewrite the Expression in Terms of 3

Finally, rewrite the expression by substituting \(3^x\) back into the expression: $$ \frac{d}{dx}(3^x) = e^{x \ln{3}} \ln{3} = 3^x \ln{3} $$ So, the derivative of \(3^x\) with respect to \(x\) is: $$ \frac{d}{dx}(3^x) = 3^x \ln{3} $$

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!

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

Use the following argument to show that \(\lim _{x \rightarrow \infty} \ln x\) \(=\infty\) and \(\lim _{x \rightarrow 0^{+}}\) \(\ln x=-\infty\). a. Make a sketch of the function \(f(x)=1 / x\) on the interval \([1,2] .\) Explain why the area of the region bounded by \(y=f(x)\) and the \(x\) -axis on [1,2] is \(\ln 2\) b. Construct a rectangle over the interval [1,2] with height \(\frac{1}{2}\) Explain why \(\ln 2>\frac{1}{2}\) c. Show that \(\ln 2^{n}>n / 2\) and \(\ln 2^{-n}<-n / 2\) d. Conclude that \(\lim _{x \rightarrow \infty} \ln x=\infty\) and \(\lim _{x \rightarrow 0^{+}} \ln x=-\infty\)

Compute the following derivatives using the method of your choice. $$\frac{d}{d x}\left(x^{2 x}\right)$$

A body of mass \(m\) is suspended by a rod of length \(L\) that pivots without friction (see figure). The mass is slowly lifted along a circular arc to a height \(h\) a. Assuming that the only force acting on the mass is the gravitational force, show that the component of this force acting along the arc of motion is \(F=m g \sin \theta\) b. Noting that an element of length along the path of the pendulum is \(d s=L d \theta,\) evaluate an integral in \(\theta\) to show that the work done in lifting the mass to a height \(h\) is \(m g h\)

The harmonic sum is \(1+\frac{1}{2}+\frac{1}{3}+\dots+\frac{1}{n} .\) Use a right Riemann sum to approximate \(\int_{1}^{n} \frac{d x}{x}(\) with unit spacing between the grid points) to show that \(1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}>\ln (n+1)\) Use this fact to conclude that \(\lim _{n \rightarrow \infty}\left(1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}\right)\) does not exist.

A spring has a restoring force given by \(F(x)=25 x .\) Let \(W(x)\) be the work required to stretch the spring from its equilibrium position \((x=0)\) to a variable distance \(x\) Graph the work function. Compare the work required to stretch the spring \(x\) units from equilibrium to the work required to compress the spring \(x\) units from equilibrium.
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Statistics

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Logic and Functions

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.