ANOTHER EXAMPLE OF THE USE OF THE DEFINITION OF A DERIVATIVE
dsinx/dx = limh --> 0[sin(x+h)-sinx]/h
sin(x + h) = sinx cosh + cosxsinh
dsinx/dx = limh --> 0[sin(x+h)-sinx]/h=
limh --> 0 [sinx cosh + cosxsinh -sinx]/h=
limh --> 0 [sinx (cosh - 1)+ cosxsinh]/h=
{limh --> 0 [cosh - 1]/h=
limh --> 0 [cosh - 1] [cosh +1]/ [cosh + 1]h=
limh --> 0 [cos2 h - 1]/ [cosh + 1]/h=
limh --> 0 -sin2h/[cosh +1]h=
limh --> 0 -sinh/h*1/[cosh +1]*sinh= -1*1/2*0=0}
limh --> 0 [sinx (cosh - 1)+ cosxsinh]/h= cosx