Question

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

Steps would be appreciated.

Answer 1

Answer:

$\frac{dy}{dx} = \frac{cos(x+y)}{1-cos(x+y)}$

Step-by-step 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  $sin (x + y)$

$\frac{dy}{dx}= cos(x+y)*\frac{d}{dx}(x+y)$

$\frac{dy}{dx}= cos(x+y)*(1+\frac{dy}{dx})\\\\\frac{dy}{dx}= cos(x+y)+\frac{dy}{dx}cos(x+y)\\\\\frac{dy}{dx} -\frac{dy}{dx}cos(x+y)=cos(x+y)\\\\\frac{dy}{dx}(1-cos(x+y))=cos(x+y)\\\\\frac{dy}{dx} = \frac{cos(x+y)}{1-cos(x+y)}$

Answer 2

Answer:

Step-by-step 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)