Question

Find dy/dx by implicit differentiation: e^y cos(x) = 1 + sin(xy)

Answer 1

Hello,

f(x,y)=e^y cos(x)-1-sin(xy)=0
@f/@x=-sin x *e^y-y*cos(xy)
@f/@y=cosx *e^y-x*sin(xy)

dy/dx=-(@f/@x)/(@f/@y)=(sin x * e^y+y*cos(xy))/(cos x *e^y-x*sin(xy)

Answer 2

Implicit expression refers to equation that are not strictly expressed in terms of y and x separately. In this case, the derivative of expression 
e^y cos(x) = 1 + sin(xy) is

-e^ y sin x dx + cos x e^y dy = cos (xy) (xdy + y dx)
dx (-e^ y sin x - y cosxy) = dy (x cos xy -cos x e^y)
dy/dx =  (-e^ y sin x - y cosxy) / (x cos xy -cos x e^y)