Question Number 84890 by Hassen_Timol last updated on 17/Mar/20
$$\mathrm{If}\:\mathrm{we}\:\mathrm{have}\::\:\:\:\:\:{y}\:=\:{e}^{{x}} \\ $$$$ \\ $$$${W}\mathrm{hat}\:\mathrm{is}\::\:\:\:\frac{\mathrm{d}}{\mathrm{d}{y}}{e}^{{x}} \:=\:… \\ $$$$ \\ $$$$\mathrm{If}\:\mathrm{we}\:\mathrm{derivate}\:\mathrm{with}\:{y}… \\ $$$$ \\ $$$$\mathrm{Please}… \\ $$
Commented by mr W last updated on 17/Mar/20
$${e}^{{x}} ={y} \\ $$$$\frac{{d}\left({e}^{{x}} \right)}{{dy}}=\frac{{d}\left({y}\right)}{{dy}}=\mathrm{1} \\ $$
Commented by Hassen_Timol last updated on 17/Mar/20
$$\mathrm{Ohhhh},\:\mathrm{it}\:\mathrm{was}\:\mathrm{really}\:\mathrm{easy}\:\mathrm{in}\:\mathrm{facts}, \\ $$$$\mathrm{I}\:\mathrm{thought}\:\mathrm{about}\:\mathrm{that}\:\mathrm{but}\:\mathrm{I}\:\mathrm{thought} \\ $$$$\mathrm{that}\:\mathrm{it}\:\mathrm{was}\:\mathrm{not}\:\mathrm{correct}\:\mathrm{since}\:\mathrm{it}'\mathrm{s}\:\mathrm{so} \\ $$$$\mathrm{easy}.\:\mathrm{Thanks}\:\mathrm{a}\:\mathrm{lot},\:\mathrm{Mr}\:\mathrm{W}. \\ $$
Commented by mr W last updated on 17/Mar/20
$${you}\:{can}\:{also}\:{do}\:{it}\:{in}\:{complex}\:{way}: \\ $$$${y}={e}^{{x}} \\ $$$$\Rightarrow{x}=\mathrm{ln}\:{y}\:\Rightarrow\frac{{dx}}{{dy}}=\frac{\mathrm{1}}{{y}}=\frac{\mathrm{1}}{{e}^{{x}} } \\ $$$$\frac{{d}\left({e}^{{x}} \right)}{{dy}}=\frac{{d}\left({e}^{{x}} \right)}{{dx}}×\frac{{dx}}{{dy}}={e}^{{x}} ×\frac{\mathrm{1}}{{y}}={e}^{{x}} ×\frac{\mathrm{1}}{{e}^{{x}} }=\mathrm{1} \\ $$