find y' using the implicit differentiation : -x +3x^2 = y^2 -3x^5*y^2

Given the functions:

-x +
3x^2 = y^2 - 3x^5 y^2

We will use the implicit
differentiation to fin y'.

(-x)' + 3(x^2)' = (y62)'
-3(x^5*y^2)'

(-x)' + 3(x^2)' = (y^2)' -3[(x^5)'*y^2 +
(x^5*(y^2)']

-1 + 6x = 2yy' - 3[ 5x^4*y^2 + x^5*
2yy']

-1 + 6x = 2yy' -15x^4*y^2 - 6x^5
*yy'

Now we will combine the terms with y' on the left
sides.

==> 6x^5*yy' - 2yy' = 1-6x -
15x^4*y^2

Now we will factor
y'.

==> y'( 6x^5 *y - 2y) = (1-6x
-15x^4*y^2)

Now we will divide by (6x^5*y -
2y)

==> y' = ( 1- 6x-15x^4*y^2) /
(6x^5 y - 2y)