/*! 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 Differentiate. $$f(x)=12+\frac... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 6

Differentiate. $$f(x)=12+\frac{1}{7^{3}}$$

Short Answer

Expert verified
The derivative is 0.

Step by step solution

01

Identify the Components of the Function

The given function is a constant function: \[ f(x) = 12 + \frac{1}{7^3} \].
02

Simplify the Function

First, simplify the constant term: \[ f(x) = 12 + \frac{1}{343} \]. Recall that \( 7^3 = 343 \).
03

Combine the Constants

Add the constants together since they are both numbers: \[ f(x) = 12 + \frac{1}{343} = 12.00291545195 \] approximately.
04

Differentiate the Constant Function

Differentiate the function. The derivative of a constant is 0: \[ f'(x) = 0 \].

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.

constant function
A constant function is one that always outputs the same value, no matter what the input is. For example, in the function \[ f(x) = 12 + \frac{1}{7^3} \], no matter what value of \( x \) you plug in, the result is always the same number. This is because there are no variables (like \( x \)) that can change.

Some characteristics of constant functions include:
  • Their graphs are horizontal lines.
  • There is no slope in a constant function's graph because there is no rate of change.
Knowing these properties can help you identify and work with constant functions easily.
simplifying constants
Simplifying constants means combining all constant terms into a single number. In the given function, \[ f(x) = 12 + \frac{1}{7^3} \], simplifying constants involves calculating \[ \frac{1}{7^3} \].

Since \[ 7^3 = 343 \], we simplify it to: \[ f(x) = 12 + \frac{1}{343} \. \] Then, you add these constants together to get a single number:
\[ f(x) = 12 + 0.00291545195 \].

Simplifying constants helps to make the function cleaner and easier to work with.
derivative of a constant
The derivative of a constant is always zero. This is because the derivative measures the rate of change of a function. Since a constant function has no change (it stays the same), its rate of change is zero.

For the given function \[ f(x) = 12 + \frac{1}{343} \], once simplified, is just a number: \[ f(x) = 12.00291545195 \].

When you differentiate this constant function, the result is: \[ f'(x) = 0 \. \] This is a fundamental rule in calculus and is easy to remember and apply.

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

Let \(y\) denote the percentage of the world population that is urban \(x\) years after 2014. According to data from the United Nations, 54 percent of the world's population was urban in 2014 , and projections show that this percentage will increase to 66 percent by 2050. Assume that \(y\) is a linear function of \(x\) since 2014. (a) Determine \(y\) as a function of \(x.\) (b) Interpret the slope as a rate of change. (c) Find the percentage of the world's population that is urban in 2020. (d) Determine the year in which \(72 \%\) of the world's population will be urban.

Use limits to compute \(f^{\prime}(x) .\) $$f(x)=\frac{1}{x^{2}+1}$$

If \(g(3)=2\) and \(g^{\prime}(3)=4,\) find \(f(3)\) and \(f^{\prime}(3),\) where \(f(x)=2 \cdot[g(x)]^{3}\).

Use a derivative routine to obtain the value of the derivative. Give the value to 5 decimal places. $$f^{\prime}(1), \text { where } f(x)=\frac{1}{1+x^{2}}$$

A ball thrown straight up into the air has height \(s(t)=102 t-16 t^{2}\) feet after \(t\) seconds. (a) Display the graphs of \(s(t)\) and \(s^{\prime}(t)\) in the window \([0,7]\) by \([-100,200] .\) Use these graphs to answer the remaining questions (b) How high is the ball after 2 seconds? (c) When, during descent, is the height 110 feet? (d) What is the velocity after 6 seconds? (e) When is the velocity 70 feet per second? (f) How fast is the ball traveling when it hits the ground?
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Decision 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.