how to differentiate ln(cos x) w.r.t x by first principles

F(x)=Log cosx

F(x+h)=log cos(x+h)

dy/dx =lim(h0)  F(x+h)-F(x) /  h

           =lim(h0)Log cos(x+h) -logcosx  /h

           =lim(h0)Log cos(x+h)/cosx /h

             =lim(h0)log(1+ cos(x+h)/cosx  -1) /h

            =lim(h0)log(1+cos(x+h)-cosx /cosx) /h

         using C and D formula of trigo ...and   lim (h0) log(1  +  x) /x  =1)

we get d/dx= -tanx