Question

Sin(x+2y)=cos(2x -y)

how would one solve this using implicit differentiation with respect to x?​

Answer 1

`~~~~~~~\sin (x+2y) = \cos (2x-y)\\\\\\\implies (d)/(dx) \sin(x+2y) = (d)/(dx) \cos (2x-y)\\\\\\\implies \cos(x+2y)(d)/(dx)(x+2y) = -\sin(2x-y) (d)/(dx)(2x-y)~~~~~~~~~~~;[\text{Chain rule}]\\\\\\\implies \cos(x+2y) \left(1+2 (dy)/(dx)\right) = -\sin(2x-y)\left(2-(dy)/(dx) \right)\\\\\\\implies \cos(x+2y) + 2\cos(x+2y)(dy)/(dx) = -2\sin(2x-y)+\sin(2x-y) (dy)/(dx)\\ \\\\`

`\implies \sin(2x-y) (dy)/(dx) - 2\cos(x+2y) (dy)/(dx) = \cos(x+2y) + 2\sin (2x-y)\\\\\\\implies \left[\sin(2x-y) -2\cos(x+2y) \right] (dy)/(dx) = \cos(x+2y) + 2\sin (2x-y)\\\\\\\implies (dy)/(dx) = (\cos(x+2y) + 2\sin (2x-y))/(\sin(2x-y) -2\cos(x+2y) )`