Question

Determine a expressao da derivada y' para a função y=fx dada implicitamente pela expresssao y³x²+13=3x²-2y+3x³

Answer 1

$y^{3}x^{2}+13=3x^{2}-2y+3x^{3}$

Derivando a expressão implicitamente em relação a x:

$\dfrac{d}{dx}(y^{3}x^{2})+\dfrac{d}{dx}(13)=\dfrac{d}{dx}(3x^{2})-\dfrac{d}{dx}(2y)+\dfrac{d}{dx}(3x^{3})\\\\\\\dfrac{d}{dx}(y^{3})\cdot x^{2}+\dfrac{d}{dx}(x^{2})\cdot y^{3}+0=3\cdot2x^{2-1}-2\cdot1y^{1-1}\dfrac{d}{dx}y+3\cdot3x^{2}\\\\\\3y^{2}x^{2}\dfrac{dy}{dx}+2x\cdot y^{3}=6x-2\dfrac{dy}{dx}+9x^{2}\\\\\\3y^{2}x^{2}y'+2xy^{3}=9x^{2}+6x-2y'$

Isolando y' (dy/dx):

$3y^{2}x^{2}y'+2xy^{3}=9x^{2}+6x-2y'\\\\3y^{2}x^{2}y'+2y'=9x^{2}+6x-2xy^{3}\\\\y'\cdot(3x^{2}y^{2}+2)=9x^{2}+6x-2xy^{3}\\\\\\\boxed{\boxed{y'=\dfrac{9x^{2}+6x-2xy^{3}}{3x^{2}y^{2}+2}}}$