/*! 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 2 Give the derivative formula for ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 2

Give the derivative formula for each function. \(\quad f(x)=5 e^{x}+3\)

Short Answer

Expert verified
The derivative of \( f(x) = 5e^x + 3 \) is \( f'(x) = 5e^x \).

Step by step solution

01

Identify the Components of f(x)

The function given is \( f(x) = 5e^x + 3 \). It consists of two components: \( 5e^x \), which involves the exponential function \( e^x \), and the constant term \( 3 \).
02

Differentiate the Exponential Component

The derivative of the exponential function \( e^x \) with respect to \( x \) is \( e^x \). So, the derivative of \( 5e^x \) will be \( 5 \) times the derivative of \( e^x \), which gives us \( 5e^x \).
03

Differentiate the Constant Component

The derivative of a constant is always zero. Therefore, the derivative of \( 3 \) is \( 0 \).
04

Combine the Derivatives

To find \( f'(x) \), we add the derivatives of each part. From Steps 2 and 3, the derivative of \( 5e^x + 3 \) is \( 5e^x + 0 \), which simplifies to \( 5e^x \).

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.

Exponential Function
An exponential function is a critical concept in calculus and mathematics. It involves a constant base raised to a variable exponent, typically expressed as \(e^x\), where \(e\) is the mathematical constant approximately equal to 2.71828. This function is special because it represents continuous growth or decay.

Some key characteristics of exponential functions include:
  • They rapidly increase or decrease in value, depending on the sign of the exponent.
  • Their derivative, or rate of change, is proportional to their current value, which makes them unique.

In the given problem \(f(x) = 5e^x + 3\), \(e^x\) represents the exponential growth, and its behavior is defined by its increasing nature. It's important to note that any constant multiplied by \(e^x\) also exhibits the same growth pattern, as seen in the term \(5e^x\).
Differentiation
Differentiation is a fundamental tool in calculus used to determine the rate at which a function is changing at any given point. This process involves calculating the derivative, which gives us a new function, representing these instantaneous rates of change.

For the expression \(f(x) = 5e^x + 3\), we apply differentiation to find \(f'(x)\):
  • The derivative of \(e^x\) is \(e^x\), thus when multiplying by a constant, the constant is retained. Therefore, derivative of \(5e^x\) becomes \(5e^x\).
  • Each term in the function is differentiated individually and results are summed to get the final derivative.

Differentiation is extremely powerful as it can be used to find slopes of curves, optimize functions, and model real-world situations, such as growth and decay processes.
Constant Term in Calculus
A constant term in calculus refers to a number without a variable, like the \(3\) in \(f(x) = 5e^x + 3\). When differentiating, the approach for a constant is straightforward, as the rate of change, or derivative, of a constant is always zero. This is because constants do not change with respect to the variable.

In practical terms, constants are static and do not contribute to the gradient of a function's graph. In our example, the derivative of \(3\) contributes \(0\) to the overall result.
  • This property simplifies differentiation tasks since any constant terms can be immediately reduced to zero.
  • It leaves only the non-constant components of a function's derivative to be determined and simplified.

Understanding how constants operate within calculus is crucial for simplifying derivatives and interpreting the constant's impact鈥攐r lack thereof鈥攐n the behavior of functions.

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

Evaluate the limit. If the limit is of an indeterminate form, indicate the form and use L'H么pital's Rule to evaluate the limit. $$ \lim _{x \rightarrow 2} \frac{2 x^{2}-5 x+2}{5 x^{2}-7 x-6} $$

Use the method of replacement or end-behavior analysis to evaluate the limits. $$ \lim _{x \rightarrow-2}\left(3 x^{2}+e^{x}\right) $$

Personal Consumption The amount spent by a consumer on nondurable goods can be modeled as \(n(x)=-1.1+1.64 \ln x\) thousand dollars and the amount spent on motor vehicles can be modeled as $$ m(x)=1.62\left(1.26^{x}\right) \text { hundred dollars } $$ where \(x\) thousand dollars is the amount spent by that same consumer for all personal consumption. (Source: Based on data from the U.S. Bureau of Labor Statistics) a. Use output values corresponding to \(\$ 4,500\) and \(\$ 10,500\) personal consumption to determine an appropriate model for personal consumption as a function of the amount spent on motor vehicles. b. Write a model giving the amount spent on nondurable goods as a function of the amount spent on motor vehicles. c. How much is spent on nondurable goods by somebody who spends \(\$ 340\) on his or her motor vehicle? At what rate is this amount changing with respect to motor vehicle spending? Write a sentence of interpretation for the results.

Airline Load Capacity The capacity of commercial large aircraft to generate revenue is measured in ton-miles. The capacity taken up by paying passengers on U.S. carriers can be modeled as $$ g(x)=16.2\left(1.009^{2.18 x+3.41}\right) \text { trillion ton-miles } $$ when a total of \(x\) trillion passenger ton-miles are flown, $$ \text { data from } 20 \leq x \leq 85 . $$ (Source: Based on data from Bureau of Transportation Statistics; applies to large certified aircraft only) a. Write a model for the rate of change of the capacity taken up by paying passengers. b. What are the capacity, the rate of change of capacity, and the percentage rate of change of capacity when 80 trillion passenger ton-miles are flown?

Write derivative formulas for the functions. $$ f(x)=\frac{2 x^{3}+3}{2.7 x+15} $$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Mechanics Maths

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.