Question

Find dy/dx if y=sin(x+y)

Steps would be appreciated.

Answer 1

Explanation:

Note that y=sin(x+y) is an implicit function; y appears on both sides of the equation, which makes it difficult or impossible to solve for y.

However, our job here is to find the derivative dy/dx.

We apply the derivative operator d/dx to both sides. Here are the results:

dy

---- = cos(x + y)(dx/dx + dy/dx), or

dx Note: dx/dx = 1

dy

---- = cos(x + y)(1 + dy/dx), or = cos(x + y) + cos(x + y)(dy/dx)

dx

We move that cos(x + y)(dy/dx) term to the left side to consolidate dy/dx terms:

dy

---- - cos(x + y)(dy/dx) = cos(x + y)

dx

or:

dy

[ ---- ] [ 1 - cos(x + y) ] = cos(x + y)

dx

Finally, we divide both sides by [ 1 - cos(x + y) ], obtaining the derivative:

dy cos(x + y)

[ ---- ] -------------------------

dx 1 - cos(x + y)

Answer 2

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

Explanation:

We have the function
`y=sin(x+y)`

We need find the derivative of y with respect to x

Note that the function
`y = sin (x + y)` depends on the variable x and the variable y. Therefore the derivative of y with respect to x will be equal to the derivative of
`sin (x + y)` by the internal derivative of
`sin (x + y)`

`(dy)/(dx)= cos(x+y)*(d)/(dx)(x+y)`

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