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Chapter 9: Problem 2

Find the differential \(d y\). \(y=3 x^{2 / 3}\)

Short Answer

Expert verified
The differential \(dy\) is given by \(dy = 2x^{-1/3} dx\).

Step by step solution

01

Understanding the Power Rule

The power rule for derivatives can be stated as follows: if \(y = x^n\), then the derivative \(\(dy/dx\) = n*x^{n-1}\). In this case, \(y = 3x^{2/3}\), so the base \(x\) is raised to the power of \(2/3\).
02

Apply the Power Rule

Applying the power rule to the function, differentiate it with respect to \(x\). That gives us \(dy/dx = (2/3) * 3x^{(2/3) - 1}\). Simplify this expression to make the calculation more precise.
03

Simplify the Expression

Simplifying the expression gives \(dy/dx = 2x^{-1/3}\).
04

Determine the Differential dy

The differential dy is found by multiplying \(dy/dx\) by \(dx\). So \(dy = 2x^{-1/3} dx\).

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Most popular questions from this chapter

Sketch the graph of the function. Choose a scale that allows all relative extrema and points of inflection to be identified on the graph. \(y=-x^{2}-2 x+3\)

Sketch the graph of the function. Label the intercepts, relative extrema, points of inflection, and asymptotes. Then state the domain of the function. \(y=\frac{x^{3}}{x^{3}-1}\)

Sketch the graph of the equation. Use intercepts, extrema, and asymptotes as sketching aids. \(y=\frac{x}{(x+1)^{2}}\)

Sketch the graph of the function. Choose a scale that allows all relative extrema and points of inflection to be identified on the graph. \(y=\left\\{\begin{array}{l}x^{2}+1, x \leq 0 \\ 1-2 x, x>0\end{array}\right.\)

Use a graphing utility to graph the function. Choose a window that allows all relative extrema and points of inflection to be identified on the graph. \(y=x^{4 / 3}\)
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