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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)
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