If we have a function, #f(x)#, then we can define the derivative of that function, #f'(x)# as:
#lim_(h->0)# #(f(x+h)-f(x))/h#
Letting #f(x)=sinx#,
#f'(x)=lim_(h->0)# #(sin(x+h)-sin(x))/h#
#=lim_(h->0)# #(sinxcos h+cosxsinh -sinx)/h#
#=lim_(h->0)# #((sinxcos h)/h) + lim_(h->0)# #((cosxsin h)/h)#
#-lim_(h->0)# #(sinx/h)#
#=sinx# #lim_(h->0)# #(cos h/h-1/h) + cosx# #lim_(h->0)# #(sin h/h)#
#lim_(h->0)# #(cos h/h-1/h)=0#
#lim_(h->0)# #(sin h/h)=1#
#thereforesinx# #lim_(h->0)# #(cos h/h-1/h) + cosx# #lim_(h->0)# #(sin h/h)#
#=sinx xx0+cosx xx1#
#=cosx#
Note: If #x# is not measured in radians, then the evaluation of the limits is different and #(sinx)'=0.0175cosx#.
Applying the same limit to #g(x)#, we eventually get:
#g'(x)= e^x# #lim_(h->0)# #((e^h-1)/h)#
#lim_(h->0)# #((a^h-1)/h)=lna#
#therefore e^x# #lim_(h->0)# #((e^h-1)/h)#
#=e^xlne#
#=e^x#