Question Number 158671 by mahdipoor last updated on 07/Nov/21
$$\forall{x}\in{R}\:;\:{f}\left({x}\right)={f}\:'\left({x}\right) \\ $$$${prove}\:{f}\left({x}+{y}\right)={f}\left({x}\right){f}\left({y}\right) \\ $$
Commented by mr W last updated on 07/Nov/21
$${generally} \\ $$$${for}\:{f}\left({x}+{y}\right)={f}\left({x}\right){f}\left({y}\right)\:{we}\:{have}\: \\ $$$${f}\left({x}\right)={a}^{{x}} \:\left({a}>\mathrm{0}\right) \\ $$$${f}'\left({x}\right)=\left(\mathrm{ln}\:{a}\right){a}^{{x}} \neq{f}\left({x}\right)\:\left({except}\:{a}={e}\right) \\ $$$${therefore} \\ $$$${f}'\left({x}\right)={f}\left({x}\right)\:{doesn}'{t}\:{mean}\: \\ $$$${f}\left({x}+{y}\right)={f}\left({x}\right){f}\left({y}\right) \\ $$$${and}\:{f}\left({x}+{y}\right)={f}\left({x}\right){f}\left({y}\right)\:{doesn}'{t}\:{mean}\: \\ $$$${f}'\left({x}\right)={f}\left({x}\right).\: \\ $$
Commented by mr W last updated on 07/Nov/21
$${you}\:{can}'{t}\:{prove}\:{because}\:{it}'{s}\:{not}\:{true}. \\ $$$${example}\:{f}\left({x}\right)=\mathrm{10}{e}^{{x}} . \\ $$$${you}\:{have}\:{f}'\left({x}\right)=\mathrm{10}{e}^{{x}} ={f}\left({x}\right) \\ $$$${but}\:{f}\left({x}+{y}\right)=\mathrm{10}{e}^{{x}+{y}} \\ $$$${f}\left({x}\right){f}\left({y}\right)=\mathrm{10}{e}^{{x}} ×\mathrm{10}{e}^{{y}} =\mathrm{100}{e}^{{x}+{y}} \\ $$$${f}\left({x}+{y}\right)\neq{f}\left({x}\right){f}\left({y}\right) \\ $$
Commented by mahdipoor last updated on 07/Nov/21
$${thanks} \\ $$