Question

dy/dx of x^2/3+y^2/3=a^2/3

Answer 1

Hey there!!!

Given : x^2/3 + y^2/3 = a^2/3

we have to find dy/dx.

Let x = a cos³Ф and y = a sin³Ф

now put the values of x and y in the left hand side of the given equation:

x^2/3 + y^2/3 = (acos³Ф)^2/3 + (asin³Ф)^2/3
                      = a^2/3 ( cos²Ф + sin²Ф )
                      = a^2/3 
Now, x = acos³Ф
∴      differentiating it w.r.t. Ф→
      dx/dФ = =3acos²Ф sinФ....................(1)

similarly,

y = a sin³Ф
differentiating it w.r.t. Ф→
dy/dФ = 3a sin²Ф cosФ..............................(2)

from the above two equations→
dy/dx = (dy/dФ) / (dx/dФ)
          = 3a sin²Ф cosФ / -3a cos²ФsinФ
          = -tanФ
          = - ∛y/x


Hence, dy/dx = -∛y/x

Hope it helps!!!