/*! 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 51 Find the indicated derivative. ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 51

Find the indicated derivative. \(\frac{d}{d x}\left(x^{3 / 4}\right)\)

Short Answer

Expert verified
\( \frac{3}{4} x^{-1/4} \)

Step by step solution

01

- Rewrite the function

Rewrite the given function to a form that is easier to differentiate. The function is already written as a power of x: \ \( x^{3/4} \).
02

- Apply the Power Rule

The Power Rule for differentiation states that if you have a function of the form \( f(x) = x^n \), then its derivative is given by \( f'(x) = nx^{n-1} \).
03

- Differentiate using the Power Rule

Set \( n = \frac{3}{4} \). Using the power rule: \ \ \[ \frac{d}{dx} \big( x^{3/4} \big) = \frac{3}{4} x^{(3/4 - 1)} \ \ \]
04

- Simplify the expression

Simplify the exponent: \ \ \[ \frac{d}{dx} \big( x^{3/4} \big) = \frac{3}{4} x^{(-1/4)} \ \ \].

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.

Power Rule
One of the most essential tools in calculus for finding derivatives is the power rule. The power rule simplifies the process of differentiation when dealing with powers of x. Specifically, if you have a function in the form of \( f(x) = x^n \), then the derivative of this function is given by \( f'(x) = nx^{n-1} \). This rule allows you to quickly and easily find the derivative without having to perform more complex calculations.
For example, consider the function \( x^{3/4} \). Using the power rule, you can set \( n = \frac{3}{4} \). Then, the derivative is \( \frac{3}{4} x^{(3/4 - 1)} \).
Derivative Calculation
Calculating derivatives involves applying rules like the power rule to break down functions into simpler parts. The derivative of a function measures how the function's output changes as its input changes. In our example, the function is \( x^{3/4} \). By applying the power rule, the derivative calculation is as follows:
  • Start with the function \( x^{3/4} \)
  • Use the power rule where \( n = \frac{3}{4} \)
  • Apply the formula to get \( \frac{d}{dx} \big( x^{3/4} \big) = \frac{3}{4} x^{(3/4 - 1)} \)
Each step helps you transform the original function into its derivative, providing insight into how the function behaves at different points.
Simplifying Exponents
After applying the power rule, it's important to simplify the exponents to get the derivative in its simplest form. Simplifying exponents makes the final expression cleaner and easier to work with. In the example, the exponent \( (3/4 - 1) \) needs to be simplified:
  • Subtract 1 from \( \frac{3}{4} \): \( \frac{3}{4} - 1 = \frac{3}{4} - \frac{4}{4} = -\frac{1}{4} \)
  • So, \( \frac{d}{dx} \big( x^{3/4} \big) = \frac{3}{4} x^{(-1/4)} \)
This result shows that the derivative of \( x^{3/4} \) is \( \frac{3}{4} x^{-1/4} \). Simplifying exponents ensures that the derivative is expressed as neatly as possible, which is crucial for further calculations and applications.

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

Estimate how much the function $$f(x)=\frac{1}{1+x^{2}}$$ will change if \(x\) decreases from 1 to \(.9 .\)

Find the indicated derivative. \(\frac{d y}{d x}\) if \(y=x^{-4}\)

Determine which of the following limits exist. Compute the limits that exist. \(\lim _{x \rightarrow 9} \frac{1}{(x-9)^{2}}\)

Based on data from the U.S. Treasury Department, the federal debt (in trillions of dollars) for the years 1995 to 2004 was given approximately by the formula $$D(x)=4.95+.402 x-.1067 x^{2}+.0124 x^{3}-.00024 x^{4}$$ where \(x\) is the number of years elapsed since the end of \(1995 .\) Estimate the federal debt at the end of \(1999(x=4)\) and the rate at which it was increasing at that time.

Let \(f(t)\) be the temperature of a cup of coffee \(t\) minutes after it has been poured. Interpret \(f(4)=120\) and \(f^{\prime}(4)=-5 .\) Estimate the temperature of the coffee after 4 minutes and 6 seconds, that is, after \(4.1\) minutes.
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Geometry

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.