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

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hala718 | High School Teacher | (Level 1) Educator Emeritus

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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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