Step 1: Plan to simplify first.
The function looks messy, but the inside of the inverse tangent can be made simple. Once simplified, differentiating becomes very easy. So we simplify before we differentiate.
Step 2: Divide top and bottom by $\cos x$.
Inside the bracket divide every term by $\cos x$. This turns $\frac{\sin x}{\cos x}$ into $\tan x$.
\[ \frac{\sin x + \cos x}{\cos x - \sin x} = \frac{\tan x + 1}{1 - \tan x} \]
Step 3: Recognise the angle-sum form.
There is a known identity $\frac{1 + \tan x}{1 - \tan x} = \tan\left(\frac{\pi}{4} + x\right)$, since $\tan\frac{\pi}{4} = 1$. So the inside is just a tangent of a sum.
Step 4: Cancel the inverse tangent.
Now $y = \tan^{-1}\left(\tan\left(\frac{\pi}{4} + x\right)\right)$. The inverse tangent undoes the tangent, leaving the angle itself.
\[ y = \frac{\pi}{4} + x \]
Step 5: Differentiate the simple result.
Now $y$ is just a constant plus $x$. The constant $\frac{\pi}{4}$ has zero derivative, and $x$ has derivative $1$.
\[ \frac{dy}{dx} = 0 + 1 = 1 \]
Step 6: State the answer.
So no matter what $x$ is, the slope is constant at $1$.
\[ \boxed{1} \]