Question

Differentiate $y=\sin (x+y)$ w.r.t $x$

Answer 1

$y=sin(x+y)$

$\frac{d}{dx}\left(y\right)=\frac{d}{dx}\left(sin\right(x+y\left)\right)$

using :

$\frac{d}{dx}\left[b\right(g\left(x\right)\left)\right]=b^{\prime }\left(g\right(x\left)\right)g^{\prime }\left(x\right)$

$=\frac{dy}{dx}=\frac{d}{dx}\left(sinx\right)(x+y)$

$=y^{\prime }=cos(x+y)\left[\frac{d}{dx}\right(x)+\frac{d}{dx}(y\left)\right]$

$=y^{\prime }cos(x+y)[1+\frac{dy}{dx}]$

$=y^{\prime }cos(x+y)[1+y^{\prime }]$

$=y^{\prime }=cos(x+y)+y^{\prime }\left(cos\right(x+y\left)\right)$

$=y^{\prime }(1-cos(x+y)=cos(x+y\left)\right)$

$=y^{\prime }=\frac{cos(x+y)}{1-cos(x+y)}$