How do you use implicit differentiation to find $\frac{d y}{d x}$ $(dy)/(dx)$ given $5 x^{3} + x y^{2} = 5 x^{3} y^{3}$ $5x^3+xy^2=5x^3y^3$ ?

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Answer 1 · 2017-01-22T20:24:58

Differentiate each term
$d (5 x^{3}) + d (x y^{2}) = d (5 x^{3} y^{3})$ $d(5x^3)+d(xy^2)=d(5x^3y^3)$
Pull constants out of each differential
$5 d (x^{3}) + d (x y^{2}) = 5 d (x^{3} y^{3})$ $5d(x^3)+d(xy^2)=5d(x^3y^3)$
Differentiate
$5 (3 x^{2} d x) + [x (2 y) d y + y^{2} d x] = 5 [x^{3} (3 y^{2}) d y + y^{3} (3 x^{2}) d x]$ $5(color(red)(3x^2dx))+[color(blue)(x(2y)dy+y^2dx)]=5[color(purple)(x^3(3y^2)dy+y^3(3x^2)dx)]$
Simplify
$(15 x^{2}) d x + (2 x y) d y + (y^{2}) d x = (15 x^{3} y^{2}) d y + (15 x^{2} y^{3}) d x$ $(15x^2)dx+(2xy)dy+(y^2)dx=(15x^3y^2)dy+(15x^2y^3)dx$
Separate $d y$ $dy$ and $d x$ $dx$ terms
$(2 x y) d y - (15 x^{3} y^{2}) d y = (15 x^{2} y^{3}) d x - (15 x^{2}) d x - (y^{2}) d x$ $(2xy)dy-(15x^3y^2)dy=(15x^2y^3)dx-(15x^2)dx-(y^2)dx$
Simplify terms
$(- 15 x^{3} y^{2} + 2 x y) d y = (15 x^{2} y^{3} - 15 x^{2} - y^{2}) d x$ $(-15x^3y^2+2xy)dy=(15x^2y^3-15x^2-y^2)dx$
Solve for $\frac{d y}{d x}$ $dy/dx$
$\frac{d y}{d x} = \frac{15 x^{2} y^{3} - 15 x^{2} - y^{2}}{- 15 x^{3} y^{2} + 2 x y}$ $dy/dx=(15x^2y^3-15x^2-y^2)/(-15x^3y^2+2xy)$

How do you use implicit differentiation to find (dy)/(dx)dydx(dy)/(dx) given 5x^3+xy^2=5x^3y^35x3+xy2=5x3y35x^3+xy^2=5x^3y^3?