#tan(x)# is differentiable where it is defined. It's undefined at #x=\pi/2+n\pi#, where #n=0,\pm 1,\pm 2,\pm 3,#... It is differentiable everywhere else.
By definition, #\tan(x)=\frac{\sin(x)}{\cos(x)}#, so as long as #\cos(x)# is not zero, #\tan(x)# is defined. Moreover, it is the ratio of two differentiable functions. The Quotient Rule says it will be differentiable wherever it is defined. Since #\cos(x)=0# when #x=\pi/2+n\pi#, where #n=0,\pm 1,\pm 2,\pm 3,#..., those are the values of #x# where #\tan(x)# fails to be differentiable. For other values of #x#, the Quotient Rule says:
#\frac{d}{dx}(\tan(x))=\frac{\cos(x)\cdot \cos(x)-\sin(x)\cdot (-\sin(x))}[\cos^{2}(x)}=\frac{1}{\cos^{2}(x)}=\sec^{2}(x).#
