#\frac{d}{dx}(\frac{x}{\cot (x)}+3\)#

Applying sum/difference rule,#\(f\pm g\)^'=f^'\pm g^'#

#=\frac{d}{dx}\(\frac{x}{\cot t(x)})+\frac{d}{dx}(3)#

Now,

#=\frac{d}{dx}\(\frac{x}{\cot t(x)})#

Applying quotient rule,#(\frac{f}{g})^'=\frac{f^'\cdot g-g^'\cdot f}{g^2}#

#=\frac{\frac{d}{dx}(x)\cot (x)-\frac{d}{dx}(\cot (x))x}{\cot ^2(x)}#

And,we know,#\frac{d}{dx}(x)=1# and, #\frac{d}{dx}(\cot (x))=-\frac{1}{\sin ^2(x)}#

Also,#\frac{d}{dx}(3)=0#

Finally,

#=\frac{\sin ^2(x)\cot(x)+x}{\sin ^2\(x)\cot ^2(x)}+0#

Simplifying it,

#\frac{x+\sin ^2t(x)\cot (x)}{\sin ^2\(x)\cot ^2\(x)}#