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Chapter 2: Problem 52

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

Short Answer

Expert verified
The second derivative \(d^{2} y / dx^{2}\) in terms of \(x\) and \(y\) is given by \[-\frac{y^{2}-2x^{2}y^{3}}{x^2 y^2}\]

Step by step solution

01

Diffrentiate

Firstly, the given equation will be differentiated with respect to \(x\) implicitly. The original equation is \(x^{2} y^{2}-2 x=3\). Differentiating one side results in the differentiation of the other side. Using the chain rule \(d(uv)=u'v+uv'\) and the power rule \(d(x^n) = n*x^{n-1}\) the first derivative is \[2x*y^{2} + x^{2}*2y*\frac{dy}{dx} -2 = 0\]
02

Isolate \(dy/dx\)

The \(\frac{dy}{dx}\) term should be isolated. Rearranging gives \[\frac{dy}{dx} = \frac{2-2x*y^{2}}{2*x^{2}*y}\]
03

Second Derivative

The second derivative, \(d^{2} y / dx^{2}\), should now be found by differentiating \(\frac{dy}{dx}\) with respect to \(x\), applying the quotient rule \(d(\frac{u}{v})=\frac{u'v - uv'}{v^2}\) gives \(d^{2} y / dx^{2}\) = \[\frac{4x^{2}y^{2}-4y^2+8x^{3}y^{3}-4x*y}{4x^{4}y^{2}}\]
04

Simplification

Now the expression can be simplified by taking common terms out and canceling others, resulting in \[\frac{-y^{2}+2x^{2}y^{3}}{x^{2}y^{2}}\]

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