Given: #25=sin(xy)/(3xy)#
Differentiate all of the terms:
#(d(25))/dx=(d(sin(xy)/(3xy)))/dx#
The derivative of a constant is 0
#0=(d(sin(xy)/(3xy)))/dx#
Use the quotient rule:
#0=(d(sin(xy)/(3xy)))/dx= ((d(sin(xy)))/dx3xy-sin(xy)(d(3xy))/dx)/(9x^2y^2)#
Using the chain rule, #(d(sin(xy)))/dx = cos(xy)(d(xy))/dx#
#0= (cos(xy)(d(xy))/dx3xy-sin(xy)(d(3xy))/dx)/(9x^2y^2)#
Use the linear property of the derivative:
#0= (cos(xy)(d(xy))/dx3xy-3sin(xy)(d(xy))/dx)/(9x^2y^2)#
Factor the #(d(xy))/dx# out of the numerator:
#0= (cos(xy)3xy-3sin(xy))/(9x^2y^2)(d(xy))/dx#
#0= (cos(xy)3xy-3sin(xy))/(9x^2y^2)(y+xdy/dx)#
Eliminate the leading coefficient by dividing it into 0:
#0= y+xdy/dx#
Subtract y from both sides
#-y=xdy/dx#
Divide both sides by x:
#dy/dx = -y/x#
