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Answer 1 · 2016-11-12T16:18:33

Although we could solve for $y$ , this looks like a classic implicit differentiation exercise.

I assume that you want the second derivative of $y$ with respect to $x$ . (Yes, there are other possibilities.)

$x^2+y^2=90$

Find $dy/dx$

$d/dx(x^2+y^2) = d/dx(90)$

$2x+2y dy/dx = 0$ , so

$dy/dx = -x/y$

Differentiate again:

$d/dx(dy/dx) = (d^2y)/dx^2 = ((-1)(y)-(-x)(dy/dx))/y^2$

Simplify a bit,

$= (-y+xdy/dx)/y^2$

Replace $dy/dx$ with its equivalent $-x/y$

$= (-y+x(-x/y))/y^2$

Simplify again

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

Clear the fraction in the nmumerator

$= ((-y-x^2/y)*y)/(y^2*y)$

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

Factor out a #-1)

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

Use the original rfelationship $x^2+y^2 = 90$ to finish simplifying.

$= (-90)/y^3$