Prove using the formal definition of a derivative that (f-g)'(x)= f'(x)-g'(x)?

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Answer 1 · 2018-02-04T09:59:41

See explanation.

According to the definition the derivative of #f(x)# is:

#f'(x)=lim_{h->0}(f(x+h)-f(x))/h#

If we apply the definition to #f(x)-g(x)# we get:

#(f-g)'(x)=lim_{h->0}([f(x+h)-g(x+h)]-[f(x)-g(x)])/h#

#=lim_{h->0}(f(x+h)-g(x+h)-f(x)+g(x))/h#

#=lim_{h->0}(f(x+h)-f(x)-(g(x+h)-g(x)))/h#

#=lim_{h->0}(f(x+h)-f(x))/h-lim_{h->0}(g(x+h)-g(x))/h#

#=f'(x)-g'(x)#

QED