Question

calcule implicitamente a derivada dy/dx em y².x³ = 12x

Answer 1

$y^2*x^3=12x\\\\ y^2=\frac{12x}{x^3}\\\\ y^2=12x^{-2}\\\\ 2y*y'=12*-2*x^{-2-1}\\\\ 2y*y'=-\frac{24}{x^3}\\\\ y'=-\frac{24}{2y*x^3}\\\\ \large\boxed{y'=-\frac{12}{y*x^3}}$

OU você pode fazer assim

$\boxed{(uv)'=u'v+uv'}$

Então:

$2y*y'*x^3+3x^2*y^2=12\\\\ 2y*x^3*y'=12-3x^2y^2\\\\ \boxed{y'=\frac{12-3x^2y^2}{2yx^3}}$

Answer 2

 Se u . v = y  , a derivada é:
 u . v' + v . u' = y'

y2 (3x²) +(x³.2yy') = 12
2x³yy'= 12 - 3x²y²
y' = (12 -3x²y²)/ (2x³y)

Logo, dy/dx = (12 -3x²y²/ (2x³y)