/*! 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 24 Find \(d y /\left.d x\right|_{x=... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 24

Find \(d y /\left.d x\right|_{x=1}\) $$ y=x^{24}+2 x^{12}+3 x^{8}+4 x^{6} $$

Short Answer

Expert verified
The value of \( \frac{dy}{dx} \) at \( x = 1 \) is 96.

Step by step solution

01

Differentiate the Function

The first step in finding \( \frac{dy}{dx} \) is to differentiate the function \( y = x^{24} + 2x^{12} + 3x^8 + 4x^6 \) with respect to \( x \). Using the power rule \( \frac{d}{dx}[x^n] = nx^{n-1} \), we find:\[ \frac{dy}{dx} = 24x^{23} + 2 \times 12x^{11} + 3 \times 8x^7 + 4 \times 6x^5. \]
02

Simplify the Derivative

Simplify the expression obtained in Step 1 by performing the multiplications:\[ \frac{dy}{dx} = 24x^{23} + 24x^{11} + 24x^7 + 24x^5. \]
03

Evaluate the Derivative at \( x = 1 \)

Substitute \( x = 1 \) into the simplified derivative:\[ \frac{dy}{dx} \bigg|_{x=1} = 24(1)^{23} + 24(1)^{11} + 24(1)^7 + 24(1)^5. \]
04

Calculate the Result

Since \( 1^n = 1 \) for any integer \( n \), the expression simplifies to:\[ 24(1) + 24(1) + 24(1) + 24(1) = 24 + 24 + 24 + 24 = 96. \]

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
The power rule is a fundamental concept in calculus, especially when dealing with derivatives of polynomial functions. It allows you to differentiate expressions of the form \( x^n \) easily. The rule states that if you have a function \( f(x) = x^n \), its derivative \( f'(x) \) is \( nx^{n-1} \).

Here鈥檚 a quick breakdown of why the power rule is so handy:
  • It simplifies the process of differentiation, making calculations faster.
  • You simply bring down the exponent as a coefficient and reduce the exponent by one.
  • It's applicable to any real number exponent \( n \), whether it's positive, negative, or a fraction.
For example, applying the power rule to \( x^{24} \) gives the derivative \( 24x^{23} \). Notice how this approach speeds up the process compared to manually computing limits.
Derivatives
Derivatives are a core concept in calculus that describe how a function changes as its input changes. Essentially, a derivative represents the rate of change or the slope of the function鈥檚 graph at any given point.

Understanding derivatives involves a few critical ideas:
  • Instantaneous Rate of Change: Unlike average rate of change, which considers over an interval, derivatives provide precise changes at a specific point.
  • Tangent Lines: The derivative at a point is also the slope of the tangent line to the function at that point.
  • General Symbol: In notation, the derivative of a function \( f \) with respect to \( x \) is represented as \( \frac{df}{dx} \).
For the function given in the exercise, calculating the derivative means finding the new function \( \frac{dy}{dx} = 24x^{23} + 24x^{11} + 24x^7 + 24x^5 \), which shows how \( y \) changes with \( x \). It鈥檚 a straightforward process, especially with polynomial functions.
Polynomial Functions
Polynomial functions are mathematical expressions that involve sums of powers of \( x \). They are versatile and appear frequently in calculus problems.

Key characteristics of polynomial functions include:
  • Defined Degrees: The degree of a polynomial function is determined by the highest power of \( x \). For example, \( x^{24} + 2x^{12} + 3x^8 + 4x^6 \) is a 24th degree polynomial.
  • Smooth and Continuous: Polynomials are smooth curves that continue infinitely in both directions without any breaks.
  • Easy to Differentiate: Differentiating polynomials is straightforward using the power rule, making them a common choice for introductory calculus exercises.
In the original exercise, the expression \( y = x^{24} + 2x^{12} + 3x^8 + 4x^6 \) represents a polynomial, and each term within it can be handled individually using calculus rules.

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

Suppose that \(L\) is the tangent line at \(x=x_{0}\) to the graph of the cubic equation \(y=a x^{3}+b x\). Find the \(x\) -coordinate of the point where \(L\) intersects the graph a second time.

Find \(d y / d x\) \(y=x^{5} \sec (1 / x)\)

Find \(d y / d x\) \(y=x^{3} \sin ^{2}(5 x)\)

Use a graphing utility to make rough estimates of the locations of all horizontal tangent lines, and then find their exact locations by differentiating. \(y=\frac{1}{3} x^{3}-\frac{3}{2} x^{2}+2 x\)

Find \(\frac{d}{d x}[f(x)]\) if \(\frac{d}{d x}[f(3 x)]=6 x\).
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

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.