/*! 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 25 Find the derivative of each of t... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 23: Problem 25

Find the derivative of each of the functions by using the definition. $$y=x^{4}-\frac{8}{x}$$

Short Answer

Expert verified
The derivative is \( y' = 4x^3 + \frac{8}{x^2} \).

Step by step solution

01

Recall the Definition of the Derivative

The definition of the derivative of a function \( f(x) \) at a point \( x \) is given by the limit: \[f'(x) = \lim_{{h \to 0}} \frac{f(x+h) - f(x)}{h}\]We will use this definition to differentiate the function \( y = x^4 - \frac{8}{x} \).
02

Substitute into the Definition

Substitute \( f(x) = x^4 - \frac{8}{x} \) into the definition of the derivative:\[y' = \lim_{{h \to 0}} \frac{(x+h)^4 - \frac{8}{x+h} - \left( x^4 - \frac{8}{x} \right)}{h}\]
03

Simplify the Expression

Expand the expression and simplify:\[(x + h)^4 = x^4 + 4x^3 h + 6x^2 h^2 + 4x h^3 + h^4\]So the expression becomes:\[y' = \lim_{{h \to 0}} \frac{x^4 + 4x^3 h + 6x^2 h^2 + 4x h^3 + h^4 - \frac{8}{x+h} - x^4 + \frac{8}{x}}{h}\]
04

Simplify the Difference Quotient Further

Combine like terms:\[y' = \lim_{{h \to 0}} \frac{4x^3 h + 6x^2 h^2 + 4x h^3 + h^4 - \left( \frac{8}{x+h} - \frac{8}{x} \right)}{h}\]Factor out \( h \) from the terms:\[y' = \lim_{{h \to 0}} \left(4x^3 + 6x^2 h + 4x h^2 + h^3 - \frac{8(x - (x+h))}{h(x+h)x}\right)\]
05

Simplify and Evaluate Limits

Since \( h \) is in the denominator, cancel it from the numerator wherever possible:\[y' = \lim_{{h \to 0}} \left(4x^3 + 6x^2 h + 4x h^2 + h^3 + \frac{8h}{h(x+h)x}\right)\]Simplify to:\[y' = \lim_{{h \to 0}} \left(4x^3 + \frac{8}{x(x+h)}\right)\]As \( h \to 0 \), terms involving \( h \) vanish and we get:\[y' = 4x^3 + \frac{8}{x^2}\]
06

Result Interpretation

We find that the derivative of \( y = x^4 - \frac{8}{x} \) is:\[ y' = 4x^3 + \frac{8}{x^2} \]This means the rate of change of the function at any \( x \) is given by \( y' \).

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.

Derivative
The derivative represents how a function changes as its input changes. It's a fundamental concept in calculus, capturing the rate of change or the slope of a curve at a given point. Calculating a derivative involves inspecting how a function behaves over an incrementally small distance. This is crucial when understanding motion, growth, or any real-world situation where change occurs continuously.
  • For instance, if you're driving a car, the speedometer shows the derivative of your position over time, which is your speed.
  • Similarly, the tangent to a curve at any point is precisely the slope given by the derivative at that point.
Understanding derivatives enables analysis and prediction of behavior across numerous fields, from physics and engineering to economics and biology.
Limit definition
The limit definition of a derivative forms the backbone of calculus. To grasp this, think about approaching a target without actually reaching it, like zooming infinitely close without contact. This concept helps us see how functions behave as they near a specific point.
  • The formal limit definition of the derivative of a function \( f(x) \) at \( x \) is expressed as: \[ f'(x) = \lim_{{h \to 0}} \frac{f(x+h) - f(x)}{h} \]
  • This definition lets us find slopes of tangent lines to the curve of a function, particularly when dealing with complex expressions not obvious to simple intuition.
Mastering this provides a powerful tool for diverse applications, letting us rigorously define how even the trickiest of functions change.
Polynomial functions
Polynomial functions are expressions composed of variables raised to whole-number exponents, combined using addition, subtraction, and multiplication. They're among the simplest types of functions and have a broad range and variety of uses.
  • For example, \( y = x^4 \) is a polynomial function of degree 4, where the highest exponent indicates the degree.
  • They are typically easy to graph and analyze due to their continuous and smooth curves.
When calculating derivatives, polynomial functions exhibit a simple pattern: each term鈥檚 derivative reduces the exponent by one and multiplies the coefficient by that original exponent. This makes them a favorite starter topic when learning calculus, as the rules are straightforward and dependable.
Rational functions
Rational functions are quotients of polynomial functions, composed of a numerator and a denominator, both being polynomials. Understanding them requires a bit more complexity than simple polynomials, as they can have discontinuities or undefined points.
  • An example is given by \( y = \frac{8}{x} \), where it's apparent that the function becomes undefined at \( x = 0 \).
  • When differentiating rational functions, employing the derivative rules requires careful handling of the quotient, often involving the product and chain rules to simplify expressions.
Rational functions are instrumental in modeling real-world behaviors in fields like engineering and economics, where quantities naturally involve divisions, such as rates, densities, and averages.

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

Solve the given problems by using implicit differentiation.Show that the graphs of \(2 x^{2}+y^{2}=24\) and \(y^{2}=8 x\) are perpendicular at the point \((2,4) .\) Display the graphs on a calculator.

In Exercises \(65-72, \lim _{x \rightarrow a^{-}} f(x)\) means to find the limit as \(x\) approaches a from the left only, and \(\lim _{x \rightarrow a^{+}} f(x)\) means to find the limit as \(x\) approaches a from the right only. These are called one-sided limits. Solve the following problems. Is there a difference between \(\lim _{x \rightarrow 2^{-}} \frac{1}{x-2}\) and \(\lim _{x \rightarrow 2^{+}} \frac{1}{x-2} ?\)

Evaluate the indicated limits by direct evaluation as in Examples \(10-14 .\) Change the form of the function where necessary. $$\lim _{h \rightarrow 0} \frac{\sqrt{9+h}-3}{h}$$

Solve the given problems by finding the appropriate derivatives.What is the instantaneous rate of change of the first derivative of \(y\) with respect to \(x\) for \(y=(1-2 x)^{4}\) for \(x=1 ?\)

Solve the given problems by finding the appropriate derivatives. In the theory of lasers, the power \(P\) radiated is given by the equation \(P=\frac{k f^{2}}{\omega^{2}-2 \omega f+f^{2}+a^{2}},\) where \(f\) is the field frequency and \(a, k,\) and \(\omega\) are constants. Find the derivative of \(P\) with respect to \(f\).
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

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.