cos (xy) = 1 + sin y

Question

calculus equation implicit differntiation.

Answer 1

Given

cos( x y ) = 1 + sin y

Differentiate each term

First term :

d / dx = cos( x y )

Second term :

Third term :

Collect terms :

cos( x y ) = 1 + sin y

   

Answer 2

The equation is cos (xy ) = 1 + sin y.

Apply derivative each side.

d [cos (xy )] = d [1 + sin y ].

- sin (xy ) d (xy ) = 0 + cos y dy.

- sin (xy ) (x dy + y dx ) = cos y dy.

- x sin (xy ) dy - y sin (xy ) dx = cos y dy

- y sin (xy ) dx = cos y dy + x sin (xy ) dy

- y sin (xy ) dx = [cos y + x sin (xy )] dy.

[- y sin (xy )] / [cos y + x sin (xy )] = dy / dx.

The solution dy / dx = [- y sin (xy )] / [cos y + x sin (xy )].