Question

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

Answer 1

`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))`