implicit differentiation

Question

e^y= sin(x+cosy)
find dy/dx using implicit differentiation

Answer 1

e^y= sin(x+cosy)

Apply derivative with respect to x each side.

(d/dx)e^y= (d/dx) [sin(x+cosy)]

e^y y'= cos(x+cosy) [(d/dx)(x+cosy)]

e^y y'= cos(x+cosy) [1- siny(y')]

y' e^y= cos(x+cosy)(1- y' siny)

y' e^y= cos(x+cosy) - y' siny cos(x+cosy)

y' e^y + y' siny cos(x+cosy) =  cos(x+cosy)

y' [ e^y + siny cos(x+cosy) ] =  cos(x+cosy)

y' =  [ cos(x+cosy) ] / [ e^y + siny cos(x+cosy) ]

Solution : y' =  [ cos(x+cosy) ] / [ e^y + siny cos(x+cosy) ]