Recall the limit definition of a derivative is:
#f'(x)-=lim_(h->0)# #((f(x+h)-f(x))/h)#
#f'(x)=5# #lim_(h->0)# #((cos(x+h)-cosx)/h)#
#=5# #lim_(h->0)# #((cosxcos h-sinxsin h-cosx)/h)#
#=5# #lim_(h->0)# #((cosxcos h-cosx-sinxsin h)/h)#
#=5# #lim_(h->0)# #((cosx(cos h-1))/h -(sinxsin h)/h)#
#=5(cosx# #lim_(h->0)# #((cos h-1)/h) -sinx# #lim_(h->0)# #(sin h/h)# #)#
#lim_(h->0)# #sin h/h=1#
#lim_(h->0)# #(cos h-1)/h=0#
#therefore5(cosx# #lim_(h->0)# #((cos h-1)/h) -sinx# #lim_(h->0)# #(sin h/h)# #)#
#=-5sinx#
#f'(x)=-5sinx#
#f'(-5)=-5sin(-5)=-4.79#
Note: the derivative of trig functions can only be evaluated as above given that #x# is in radians.
