Question

calcule a derivada implicita dx/dy e dy/dx
a) (2x + 3y)^3 + 2xy - 5x + 6y = 19xy^2
b) xcosy + 2ysen3x - 3xy = 2x^2y + 6y

Answer 1

Resposta:

a) (2x + 3y)^3 + 2xy - 5x + 6y = 19xy^2

8x³+3*4x²*3y+3*2x*9y²+9y³+ 2xy - 5x + 6y = 19xy^2

8x³+36x²y+54x*y³+9y³+ 2xy - 5x + 6y = 19xy^2

24x²+72xy+36x²dy/dx+54y³+54x3y²*dy/dx+27y²*dy/dx+2y+2xdy/dx-5+6dy/dx=19y²+38xy*dy/dx

24x²+72xy+36x²dy/dx+54y³+162x*y²*dy/dx+27y²*dy/dx+2y+2xdy/dx-5+6dy/dx=19y²+38xy*dy/dx

24x²+72xy+54y³+2y-5-19y² =-36x²dy/dx-162x*y²*dy/dx-27y²*dy/dx-2xdy/dx-6dy/dx+38xy*dy/dx

24x²+72xy+54y³+2y-5-19y² =(-36x²-162x*y²-27y²-2x-6+38xy)*dy/dx

dy/dx =(24x²+72xy+54y³+2y-5-19y²)/(-36x²-162x*y²-27y²-2x-6+38xy)

b)

xcosy + 2ysen3x - 3xy = 2x^2*y + 6y  

cosy + x *(-seny)*dy/dx+2*sen3x+(2y*3*cos(3x))+2 *dy/dx * sen3x-3y -3x*dy/dx= 2*2x*y+2*x²*dy/dx+6dy/dx

cosy +2*sen3x+(2y*3*cos(3x))+ * sen3x-3y - 2*2x*y =-x *(-seny)*dy/dx-2*sen3x *dy/dx +3x*dy/dx+2*x²*dy/dx+6dy/dx

(cosy +2*sen3x+(2y*3*cos(3x))+ * sen3x-3y - 2*2x*y) =(-x *(-seny)-2*sen3x  +3x+2*x²+6)dy/dx

dy/dx=(cosy +2*sen3x+(2y*3*cos(3x))+ * sen3x-3y - 2*2x*y)/(-x *(-seny)-2*sen3x  +3x+2*x²+6)