Question

Determine dy/dx por derivação implícita:
a) lnxy+x+y=2

Answer 1

ln (x*y) + x + y  = 2

d [ln (x*y) + x + y ]/dx = d 2/dx

(x*y)' /(x*y) + 1 + dy/dx =0

(x'*y + x* dy/dx)  *  (1/xy) + 1 +dy/dx =0

y/xy + (x/(xy) ) * dy/dx  + 1 + dy/dx =0

1/x +1/y  * dy/dx + 1 + dy/dx =0

dy/dx *(1/y + 1) = -1 - 1/x

dy/dx *(1+y)/y =-(x-1)/x

dy/dx = -y(x-1)/x(1+y)

Answer 2

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$\sf \ell n(xy)+x+y=2\\\sf \ell n(x)+\ell n(y)+x+y=2\\\sf\dfrac{1}{x}+\dfrac{1}{y}\cdot\dfrac{dy}{dx}+1+\dfrac{dy}{dx}=0\cdot (xy)\\\sf y+x\dfrac{dy}{dx}+xy+xy\dfrac{dy}{dx}=0\\\sf x\dfrac{dy}{dx}+xy\dfrac{dy}{dx}=-xy-y\\\sf\dfrac{dy}{dx}(x+xy)=-xy-y\\\sf \dfrac{dy}{dx}=-\dfrac{xy+y}{x+xy}\blue{\checkmark}$