How do you find (d^2y)/(dx^2) for 5y^2+2=5x^2?

2 Answers
Nov 25, 2016

(d^2y)/dx^2=(y^2-x^2)/y^3

Explanation:

To find (d^2y)/(dx^2) is to find the second differentiation of the
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given expression .
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5y^2+2=5x^2
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d/dx(5y^2+2)=d/dx(5x^2)
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rArr10y(dy/dx)+0=10x
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rArr10y(dy/dx)=10x
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rArrdy/dx=(10x)/(10y)
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Therefore,
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color(blue)(dy/dx=x/y
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Let us compute (d^2y)/dx^2
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(d^2y)/dx^2=d/dx(dy/dx)
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(d^2y)/dx^2=d/dx(x/y)
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Differentiating x/y is determined by applying the quotient rule differentiation.
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(d^2y)/dx^2=((dx/dx)xxy-(dy/dx)xx x)/y^2
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(d^2y)/dx^2=(y-xcolor(blue)(dy/dx))/y^2
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(d^2y)/dx^2=(y-x(x/y))/y^2
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(d^2y)/dx^2=(y-x^2/y)/y^2
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Therefore,
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(d^2y)/dx^2=(y^2-x^2)/y^3

Nov 25, 2016

The answer is =-2/5((5x^2-2)/5)^(-3/2)

Explanation:

Let's do the implicit differentiation

5y^2+2=5x^2

10ydy/dx+0=10x

dy/dx=x/y

This is of the form u/v

(u/v)'=(u'v-uv')/v^2

So, (d^2y)/dx^2=(x/y)'=(y-xdy/dx)/y^2

=(y-x^2/y)/y^2=(y^2-x^2)/y^3

5y^2=5x^2-2

y^2=(5x^2-2)/5

y^3=((5x^2-2)/5)^(3/2)

Therefore, (d^2y)/dx^2=((5x^2-2)/5-x^2)/((5x^2-2)/5)^(3/2)

=-2/(5((5x^2-2)/5)^(3/2))

=-2/5((5x^2-2)/5)^(-3/2)