Question

what is the derivative of cos(x+y)

Answer 1

$$\begin{gathered} \frac{d}{dx}\left[\cos \left(x+y\right)\right] \\ =\frac{d}{dx}\left[\cos x\cos y-\sin x\sin y\right] \end{gathered}$$
$$\begin{gathered} =\cos y(-\sin x)+\cos x(-\sin y)\frac{dy}{dx}-\sin y\cos x-\sin x\cos y\frac{dy}{dx} \\ =-\left(\sin x\cos y+\sin y\cos x\right)-\frac{dy}{dx}\left(\sin x\cos y+\sin y\cos x\right) \\ =-\left(\sin x\cos y+\sin y\cos x\right)\left(1+\frac{dy}{dx}\right) \end{gathered}$$
$=-\sin \left(x+y\right)\left(1+\frac{dy}{dx}\right)$