Question Number 80442 by Power last updated on 03/Feb/20
Commented by john santu last updated on 03/Feb/20
$${f}\:'\left({a}\right)\:{hahaha} \\ $$
Commented by Power last updated on 03/Feb/20
$$\mathrm{solution}\:\mathrm{sir}\:\mathrm{pls} \\ $$
Commented by mr W last updated on 03/Feb/20
$${let}\:{a}={x}+\Delta{h} \\ $$$$\underset{{x}\rightarrow{a}} {\mathrm{lim}}\frac{{f}\left({x}\right)−{f}\left({a}\right)}{{x}−{a}} \\ $$$$=\underset{{x}\rightarrow{a}} {\mathrm{lim}}\frac{{f}\left({a}\right)−{f}\left({x}\right)}{{a}−{x}} \\ $$$$=\underset{\Delta{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{f}\left({x}+\Delta{h}\right)−{f}\left({x}\right)}{\Delta{h}}\:\Rightarrow\:{definition}\:{of}\:\frac{{df}\left({x}\right)}{{dx}} \\ $$$$={f}'\left({x}\right)\:\left({or}\:={f}'\left({a}\right)\:{in}\:{same}\:{way}\right) \\ $$
Commented by msup trace by abdo last updated on 03/Feb/20
$${if}\:{f}\:{is}\:{not}\:{derivable}\:{at}\:{a}\:{this} \\ $$$${limit}\:{dont}\:{exist}\:{or}\:{infinite}...! \\ $$
Answered by mind is power last updated on 03/Feb/20
$${nothing}\:{too}\:{say}\: \\ $$$${if}\:{f}\left({x}\right)=\mid{x}−{a}\mid\:{we}\:{get}\:\underset{−} {+}\mathrm{1}\:\:{depend}\:{in}\:{how}\:{x}\rightarrow{a} \\ $$$${if}\:{f}\:{is}\:{differentiable}\:{arround}\:{a} \\ $$$$\Rightarrow{f}\left({x}\right)={f}\left({a}\right)+{f}'\left({a}\right)\left({x}−{a}\right)+{o}\left({x}−{a}\right) \\ $$$$\Rightarrow\frac{{f}\left({x}\right)−{f}\left({a}\right)}{{x}−{a}}={f}'\left({a}\right)+{o}\left(\mathrm{1}\right) \\ $$$$\Rightarrow\underset{{x}\rightarrow{a}} {\mathrm{lim}}\frac{{f}\left({x}\right)−{f}\left({a}\right)}{{x}−{a}}={f}'\left({a}\right) \\ $$
Commented by Power last updated on 03/Feb/20
$$\mathrm{thanks} \\ $$