/*! 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 20 Find \(\frac{d y}{d x}\). $$y=... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 20

Find \(\frac{d y}{d x}\). $$y=-4.8 x^{1 / 3}$$

Short Answer

Expert verified
-1.6x^{-2/3}

Step by step solution

01

Understand the Function

The given function is \[y = -4.8 x^{1/3}\]. This is a power function where the exponent is \frac{1}{3}.
02

Apply the Power Rule

To differentiate a function of the form \[y = ax^n\], use the power rule: \frac{d}{d x} (ax^n) = a \cdot n \cdot x^{n-1}. Here, \ a = -4.8 \ and \ n = \frac{1}{3}\.
03

Differentiate the Function

Using the power rule:\[\frac{d y}{d x} = -4.8 \cdot \frac{1}{3} \cdot x^{1/3 - 1}\]. Simplify the exponent first: \[1/3 - 1 = 1/3 - 3/3 = -2/3.\].
04

Simplify the Derivative

Now, multiply the constants and write the final expression: \[\frac{d y}{d x} = -4.8 \cdot \frac{1}{3} \cdot x^{-2/3} = \-1.6 x^{-2/3}\].

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. It helps us differentiate functions of the form \[ ax^n \]. This rule states: \[ \frac{d}{d x} (ax^n) = a \cdot n \cdot x^{n-1} \]. Breaking this down: \[ 'a' \] is a constant multiplied by the variable \[ x \], \[ 'n' \] is the exponent, and \[ x^{n-1} \] is the new exponent after differentiating.

For instance, if we have \[ y = -4.8 x^{1/3} \], we identify that \[ a = -4.8 \] and \[ n = \frac{1}{3} \]. Using the power rule, we:
  • Multiply the original coefficient (\[ -4.8 \]) by the exponent (\[ \frac{1}{3} \]).
  • Subtract one from the exponent (\[ \frac{1}{3} - 1 \]) to get the new exponent.

    This results in the derivative: \[ \frac{d}{dx}(-4.8 x^{1 / 3}) = -4.8 \cdot \frac{1}{3} \cdot x^{-2/3} \]
Exponent
Exponents are a shorthand way to represent repeated multiplication of a number by itself. In our function \[ y = -4.8 x^{1/3} \], the exponent is \[ \frac{1}{3} \], which means we're dealing with the cube root of \[ x \].

When differentiating, we need to adjust the exponent: \[ n - 1 \]. For \[ n = \frac{1}{3} \], subtracting 1 gives us:
  • \[ \frac{1}{3} - 1 = \frac{1}{3} - \frac{3}{3} = -\frac{2}{3} \].
This new exponent tells us how the rate of change of \[ y \] with respect to \[ x \] declines, given as \[ x^{-2/3} \].

Understanding exponents is crucial when applying differentiation rules and interpreting the results.
Simplification
Simplification helps in making our derivatives more readable. After applying the power rule, sometimes we end up with expressions that are cumbersome.

In the step-by-step solution, you'll notice we simplify: \[ -4.8 \cdot \frac{1}{3} = -1.6 \]. This scalar multiplication makes the function easier to discuss and apply.

Additionally, managing exponents:
  • \[ x^{-2/3} \] is simpler than its expanded form \[ x^{ \left(\frac{-2}{3}\right)} \].
  • Visualization and practical handling becomes straightforward.


Simplification ensures our final answer (\

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

Cruzin' Boards has found that the cost, in dollars, of producing \(x\) skateboards is given by \(C(x)=900+18 x^{0.7}\) If the revenue from the sale of \(x\) skateboards is given by \(R(x)=75 x^{0.8},\) find the rate at which the average profit per skateboard is changing when 20 skateboards have been built and sold.

If \(f(x)\) is a function, then \((f \circ f)(x)=f(f(x))\) is the composition of \(f\) with itself. This is called an iterated function, and the composition can be repeated many times. For example, \((f \circ f \circ f)(x)=f(f(f(x))) .\) Iterated functions are very useful in many areas, including finance (compound interest is \(a\) simple case) and the sciences (in weather forecasting, for example). For the each function, use the Chain Rule to find the derivative.. If \(f(x)=x+\sqrt{x},\) find \(\frac{d}{d x}[(f \circ f)(x)]\).

Use the Chain Rule to differentiate each function. You may need to apply the rule more than once. $$f(x)=\left(2 x^{5}+(4 x-5)^{2}\right)^{6}$$

Summertime Fabrics finds that the cost, in dollars, of producing \(x\) jackets is given by \(C(x)=950+15 \sqrt{x} .\) Find the rate at which the average cost is changing when 400 jackets have been produced.

The following is the beginning of an alternative proof of the Quotient Rule that uses the Product Rule and the Power Rule. Complete the proof, giving reasons for each step. Proof. Let $$Q(x)=\frac{N(x)}{D(x)}$$ Then $$Q(x)=N(x) \cdot[D(x)]^{-1}$$ Therefore,
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

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.