Question

Question #ce2e0

Answer 1

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

Explanation:

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`