# What is the implicit derivative of #y=x^3y+x^2y^4-5y #?

Let's first put all our variables on one side.

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To find the implicit derivative of ( y = x^3y + x^2y^4 - 5y ), differentiate both sides of the equation with respect to ( x ), treating ( y ) as a function of ( x ) using the product rule and chain rule as needed.

[ \frac{{dy}}{{dx}} = \frac{{d}}{{dx}}(x^3y) + \frac{{d}}{{dx}}(x^2y^4) - \frac{{d}}{{dx}}(5y) ]

Apply the product rule and chain rule as necessary:

[ \frac{{dy}}{{dx}} = 3x^2y + x^3\frac{{dy}}{{dx}} + 2xy^4 + x^2(4y^3)\frac{{dy}}{{dx}} - 5\frac{{dy}}{{dx}} ]

Rearrange terms to isolate ( \frac{{dy}}{{dx}} ):

[ \frac{{dy}}{{dx}} - x^3\frac{{dy}}{{dx}} - 4x^2y^4\frac{{dy}}{{dx}} = 3x^2y + 2xy^4 - 5y ]

Factor out ( \frac{{dy}}{{dx}} ):

[ \left(1 - x^3 - 4x^2y^4\right)\frac{{dy}}{{dx}} = 3x^2y + 2xy^4 - 5y ]

Divide both sides by ( 1 - x^3 - 4x^2y^4 ) to solve for ( \frac{{dy}}{{dx}} ):

[ \frac{{dy}}{{dx}} = \frac{{3x^2y + 2xy^4 - 5y}}{{1 - x^3 - 4x^2y^4}} ]

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