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Chapter 2: Problem 55

Find \(d^{2} y / d x^{2}\) in terms of \(x\) and \(y\). $$ y^{2}=x^{3} $$

Short Answer

Expert verified
\(\frac{d^2y}{dx^2}=\frac{3x^2-3x^4/y^2}{4y^2}\)

Step by step solution

01

Differentiate Both Sides Respect to \(x\)

First, take the derivative of both sides of the equation. Using the power rule \(\frac{d}{dx}(x^n)=nx^{n-1}\), for left side, \(\frac{d}{dx}(y^2)=2y\frac{dy}{dx}\), and for right side, \(\frac{d}{dx}(x^3)=3x^2\). Therefore, we get this first derivative: \(2y\frac{dy}{dx}=3x^2\).
02

Solve for \(\frac{dy}{dx}\)

To isolate \(\frac{dy}{dx}\), divide both sides of the equation by \(2y\). This will give us the first derivative \(\frac{dy}{dx}=\frac{3x^2}{2y}\).
03

Second Derivative Calculation

To find \(\frac{d^2y}{dx^2}\) we need to take the derivative of \(\frac{dy}{dx}\) again. Using the quotient rule, which states that \(\frac{d}{dx}(\frac{u}{v})= \frac{vu'-uv'}{v^2}\), where \(u=3x^2\) and \(v=2y\), we have \(u'=\frac{d}{dx}(3x^2)=6x\), and \(v'=\frac{d}{dx}(2y)=2\frac{dy}{dx}\). Substituting values, we get \(\frac{d^2y}{dx^2}=\frac{2y(6x)-3x^2 \cdot 2\frac{dy}{dx}}{(2y)^2}\).
04

Substitute the value of \(\frac{dy}{dx}\)

Substitute \(\frac{dy}{dx}=\frac{3x^2}{2y}\) into the equation and simplify. The final answer is \(\frac{d^2y}{dx^2}=\frac{3x^2-3x^4/y^2}{4y^2}\).

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Most popular questions from this chapter

In Exercises \(75-80\), evaluate the derivative of the function at the indicated point. Use a graphing utility to verify your result. \(\frac{\text { Function }}{y=\sqrt[5]{3 x^{3}+4 x}} \quad \frac{\text { Point }}{(2,2)}\)

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