/*! 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 1 Find the differential \(d y\). ... [FREE SOLUTION] | 91影视

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Chapter 9: Problem 1

Find the differential \(d y\). \(y=3 x^{2}-4\)

Short Answer

Expert verified
The differential \(d y\) of the function \(y = 3x^{2} - 4\) is \(dy = 6x\).

Step by step solution

01

Identify the function

The function given in this problem is \(y = 3x^{2} - 4\). We are asked find the differential \((dy)\) of the function.
02

Understand the definition

The differential of a function, noted as \(dy\), is equal to the derivative of the function (\(y'\)) with respect to \(x\) times \(dx\). This means, if \(y = f(x)\), then \(dy = f'(x) \cdot dx\). In this exercise, \(dx\) is understood to be 1, so differential \(dy\) will be equivalent to derivative \(f'(x)\).
03

Differentiate the function

Differentiate the function \(y = 3x^{2} - 4\) with respect to \(x\). To do so, apply the power rule for differentiation, which states that the derivative of \(x^n\) is \(n \cdot x^{n-1}\). So for the first term, \(3x^{2}\), the derivative is \(2 \cdot 3x^{2-1}\), or \(6x\). Since the derivative of a constant is 0, the derivative of \(-4\) is 0. Therefore, \(dy = 6x - 0 = 6x\).

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Most popular questions from this chapter

The demand function for a product is modeled by \(p=75-0.25 x\) (a) If \(x\) changes from 7 to 8 , what is the corresponding change in \(p\) ? Compare the values of \(\Delta p\) and \(d p\). (b) Repeat part (a) when \(x\) changes from 70 to 71 units.

Sketch the graph of the function. Choose a scale that allows all relative extrema and points of inflection to be identified on the graph. \(y=3 x^{4}+4 x^{3}\)

Sketch the graph of the function. Label the intercepts, relative extrema, points of inflection, and asymptotes. Then state the domain of the function. \(y=\frac{x^{2}-6 x+12}{x-4}\)

Sketch the graph of the function. Choose a scale that allows all relative extrema and points of inflection to be identified on the graph. \(y=x^{4}-4 x^{3}+16 x\)

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. \(y=\frac{x}{x^{2}+1}\)
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