Question

How do you find the derivative of `y = sin(x+y)`?

Answer 1

`dy/dx= cos(x+y)/{1-cos(x+y)}`

Explanation:

You simply differentiate both sides with respect to `x`. The left side would simply give you `dy/dx`. For the right side, however, you must make use of the chain rule for derivatives of composite functions (functions of functions). Thus

`d/dx (sin(x+y)) = cos(x+y) xx d/dx (x+y) = cos(x+y) (1+dy/dx)`

Thus, we get

`dy/dx = cos(x+y) (1+dy/dx)`

We can easily solve this for the quantity `dy/dx`:

`(1-(cos(x+y)) dy/dx = cos(x+y) implies dy/dx= cos(x+y)/{1-cos(x+y)}`