/*! 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 56 Find \(d^{2} y / d x^{2}\) in te... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 56

Find \(d^{2} y / d x^{2}\) in terms of \(x\) and \(y\). $$ y^{2}=4 x $$

Short Answer

Expert verified
The result of the second derivative is \(-4/y^{3}\).

Step by step solution

01

Apply 1st Implicit Differentiation

To begin, differentiate both sides of the equation \(y^{2} = 4x\) with respect to \(x\) using the chain rule. As \(y\) is also a function of \(x\), their relation is implicit. Thus, when differentiating \(y^{2}\) the chain rule gives \(2y\frac{dy}{dx}\). Differentiating \(4x\) gives \(4\). The first order derivative equation becomes: \[2y \frac{dy}{dx} = 4\].
02

Solve for First Derivative

Next, we need to solve for \(\frac{dy}{dx}\). This can be done by dividing by \(2y\) at both sides of equation to isolate \(\frac{dy}{dx}\) at one side of the equation. When we do this, the solution for \(\frac{dy}{dx}\) is found to be \[\frac{dy}{dx} = \frac{4}{2y} = \frac{2}{y}\].
03

Apply 2nd Implicit Differentiation

In order to find the second derivative, we must take the derivative of the first derivative \(\frac{dy}{dx}= \frac{2}{y}\) with respect to \(x\). Apply the chain rule again for the second implicit differentiation: \[\frac{d^{2}y}{dx^{2}} = \frac{d}{dx}\Big(\frac{2}{y}\Big) = -\frac{2}{y^{2}}\frac{dy}{dx}\]
04

Substitution

Now replace \(\frac{dy}{dx}\) in equation with \(\frac{2}{y}\) from step 2: \[\frac{d^{2}y}{dx^{2}} = -\frac{2}{y^{2}} \cdot \frac{2}{y} = -\frac{4}{y^{3}}\]

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Most popular questions from this chapter

In Exercises 107-110, (a) use a graphing utility to find the derivative of the function at the given point, (b) find an equation of the tangent line to the graph of the function at the given point, and (c) use the utility to graph the function and its tangent line in the same viewing window. \(y=\left(t^{2}-9\right) \sqrt{t+2}, \quad(2,-10)\)

Determine the point(s) at which the graph of \(f(x)=\frac{x}{\sqrt{2 x-1}}\) has a horizontal tangent line.

In Exercises 43 and 44, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The slope of the graph of the inverse tangent function is positive for all \(x\).

In Exercises \(81-88\), (a) find an equation of the tangent line to the graph of \(f\) at the indicated point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the derivative feature of a graphing utility to confirm your results. \(\frac{\text { Function }}{y=\cos 3 x} \quad \frac{\text { Point }}{\left(\frac{\pi}{4},-\frac{\sqrt{2}}{2}\right)}\)

In Exercises 103 and \(104,\) the relationship between \(f\) and \(g\) is given. Explain the relationship between \(f^{\prime}\) and \(g^{\prime}\). \(g(x)=f\left(x^{2}\right)\)
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