How do you find the derivative of #y = 2x+ cos(xy) #?

1 Answer
Dec 21, 2015

#dy/dx = (2- y sin(xy))/(1+sin(xy)) #

Explanation:

#y=2x+cos(xy)#
Differentiate with respect to x on both the sides.

#dy/dx = (d(2x))/dx + (d(cos(xy)))/dx#
#dy/dx = 2 dx/dx-sin(xy) * (d(xy))/dx # Chain rule
#dy/dx = 2 - sin(xy) * {x*dy/dx + y dx/dx}# Product rule.
#dy/dx = 2-sin(xy)*{x dy/dx + y} #
#dy/dx = 2 -x sin(xy) dy/dx - y*sin(xy) #
#dy/dx + x sin(xy) dy/dx) = 2 - y sin(xy) # collecting dy/dx together on one side of the equation.
#(1+x sin(xy)) dy/dx = 2 - y sin(xy) #

To solve for dy/dx we divide both sides by #(1+ x sin(xy))#

We get #dy/dx = (2- y sin(xy))/(1+sin(xy)) #