How do you find the derivative of #7=xy-e^(xy)#?

1 Answer
Sep 18, 2016

#dy/dx = -y/x#

Explanation:

By implicit differentiation:

#d/dx(7) = d/dx(xy - e^(xy))#

#d/dx(7) = d/dx(xy) + d/dx(e^(xy))#

By using the identity #d/dx(e^(b(x))) = (b(x))' xx e^(b(x))#

#0 = y + x(dy/dx) + (y + x(dy/dx))e^(xy)#

#0 = y + x(dy/dx) + ye^(xy) + xe^(xy)(dy/dx)#

#-y - ye^(xy) = dy/dx(x + xe^(xy))#

#(-y - ye^(xy))/(x + xe^(xy)) = dy/dx#

#-(y + ye^(xy))/(x + xe^(xy)) = dy/dx#

#dy/dx = -(y(1 + e^(xy)))/(x(1 + e^(xy)))#

#dy/dx = -y/x#

Hopefully this helps!