How do you find $\frac{d y}{d x}$ $(dy)/(dx)$ given $- 4 x^{2} y^{3} + 2 = 5 x^{2} + y^{2}$ $-4x^2y^3+2=5x^2+y^2$ ?

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Answer 1 · 2016-10-04T22:25:19

$y'=(8xy^3+10x)/(-4x^2-3y^2-2y)$

Remember that Implicit Differentiation is really just a special case of the Chain Rule.

Every time that we differentiate the a factor or term what includes the variable y we have to include a factor of $dy/dx$ or $y'$ .

For the first term, $-4x^2y^3$ , we have to use the Product Rule and Power Rule .
For the constant, $2$ , we have to use the Constant Rule .
For the term, $5x^2$ , use the Power Rule .
For the term, $y^2$ , use the Power Rule .

$-4x^2 3y^2y'+(-8)xy^3+0=10x+2yy'$

Gather the terms with $y'$ on one side of the equations and other terms on the other side.

$-4x^2 3y^2y'-2yy'=8xy^3+10x$

Factor out $y'$

$y'(-4x^2 3y^2-2y)=8xy^3+10x$

Isolate $y'$ by dividing both sides by $(-4x^2 3y^2-2y)$

$y'cancel(-4x^2 3y^2-2y)/cancel(-4x^2 3y^2-2y)=(8xy^3+10x)/(-4x^2 3y^2-2y)$

$y'=(8xy^3+10x)/(-4x^2 3y^2-2y)$

How do you find (dy)/(dx)dydx(dy)/(dx) given -4x^2y^3+2=5x^2+y^2−4x2y3+2=5x2+y2-4x^2y^3+2=5x^2+y^2?