find dy/dx if y/x^3 + x/y^3 = x^2 y^4

Given the equation:

y/x^3 +
x/y^3 = x^2*y^4

We need to find
dy/dx

We will use implicit
differentiation.

First we will multiply both sides by
x^3*y^3

==> y^4 + x^4 = x^2 * y^4 * x^3 *
y^3

But we know that x^a *x^b =
x^(a+b)

==> y^4 + x^4 = x^5*
y^7

Now we will
differentiate.

==> 4y^3*y' + 4x^3 = (x^5)'*y^7 +
x^5*(y^7)'

==> 4y^3 y'+ 4x^3 = 5x^4 y^7 + 7x^5*y^6
*y'

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

==> 4y^3 y' - 7x^5 y^6 y' = 5x^4 y^7 -
4x^3

Now we will factor
y'.

==> y' (4y^3 - 7x^5 y^6) = (5x^4 y^7 -
4x^3)

Now we will divide
.

==> y' = ( 5x^4 y^7 - 4x^3) / (4y^3 - 7x^5
y^6)

==> y' =  x^3 ( 5xy^7 -4)/ y^3 (
4 - 7x^5 y^3)