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Question Number 215918 by issac last updated on 21/Jan/25
why ((d )/dz)e^z =e^z ?? why dose not (de^z /dz)=(de^z /dz)=(e^z /z) ???
$$\mathrm{why}\:\:\frac{\mathrm{d}\:\:}{\mathrm{d}{z}}{e}^{{z}} ={e}^{{z}} \:?? \\ $$$$\mathrm{why}\:\mathrm{dose}\:\mathrm{not}\:\frac{\mathrm{d}{e}^{{z}} }{\mathrm{d}{z}}=\frac{\cancel{\mathrm{d}}{e}^{{z}} }{\cancel{\mathrm{d}}{z}}=\frac{{e}^{{z}} }{{z}}\:??? \\ $$
Answered by Frix last updated on 21/Jan/25
You mean generally (d/dx)f(x)=((f(x))/x)?
$$\mathrm{You}\:\mathrm{mean}\:\mathrm{generally}\:\frac{{d}}{{dx}}{f}\left({x}\right)=\frac{{f}\left({x}\right)}{{x}}? \\ $$
Answered by mahdipoor last updated on 21/Jan/25
The symbol d here is an operator : (d/dz)e^z ≡((e^(z+Δz) −e^z )/(Δz)) when (Δz→0) whose name is derivative not mean : (d/dz)e^z =((d×e^z )/(d×z))
$$ \\ $$$$\mathrm{The}\:\mathrm{symbol}\:{d}\:\mathrm{here}\:\mathrm{is}\:\mathrm{an}\:\mathrm{operator}\:: \\ $$$$\frac{\mathrm{d}}{\mathrm{d}{z}}{e}^{{z}} \equiv\frac{{e}^{{z}+\Delta{z}} −{e}^{{z}} }{\Delta{z}}\:\:{when}\:\left(\Delta{z}\rightarrow\mathrm{0}\right) \\ $$$$\mathrm{whose}\:\mathrm{name}\:\mathrm{is}\:\mathrm{derivative} \\ $$$${not}\:{mean}\:: \\ $$$$\frac{{d}}{{dz}}{e}^{{z}} =\frac{{d}×{e}^{{z}} }{{d}×{z}} \\ $$
Answered by MathematicalUser2357 last updated on 29/Mar/25
You maybe forgot the notation of derivatives. The derivative notations are: (dy/dx), (d/dx)y, y′, etc. (Derivative notation is marked blue) Solving it with your derivative skills with that notation you forgot: (d/dz) e^z =e^z . If you didn′t forgot the notation of derivatives, You maybe forgot the first principle of derivatives. If you didn′t forgot that, you thinked (d/dz)e^z as an Algebra question. Please, let that notation go to your mind.
$$\mathrm{You}\:\mathrm{maybe}\:\mathrm{forgot}\:\mathrm{the}\:\mathrm{notation}\:\mathrm{of}\:\mathrm{derivatives}. \\ $$$$\mathrm{The}\:\mathrm{derivative}\:\mathrm{notations}\:\mathrm{are}:\:\frac{{dy}}{{dx}},\:\frac{{d}}{{dx}}{y},\:{y}',\:\mathrm{etc}.\:\left(\mathrm{Derivative}\:\mathrm{notation}\:\mathrm{is}\:\mathrm{marked}\:\mathrm{blue}\right) \\ $$$$\mathrm{Solving}\:\mathrm{it}\:\mathrm{with}\:\mathrm{your}\:\mathrm{derivative}\:\mathrm{skills}\:\mathrm{with}\:\mathrm{that}\:\mathrm{notation}\:\mathrm{you}\:\mathrm{forgot}: \\ $$$$\frac{{d}}{{dz}}\:{e}^{{z}} ={e}^{{z}} . \\ $$$$\mathrm{If}\:\mathrm{you}\:\mathrm{didn}'\mathrm{t}\:\mathrm{forgot}\:\mathrm{the}\:\mathrm{notation}\:\mathrm{of}\:\mathrm{derivatives}, \\ $$$$\mathrm{You}\:\mathrm{maybe}\:\mathrm{forgot}\:\mathrm{the}\:\mathrm{first}\:\mathrm{principle}\:\mathrm{of}\:\mathrm{derivatives}. \\ $$$$\mathrm{If}\:\mathrm{you}\:\mathrm{didn}'\mathrm{t}\:\mathrm{forgot}\:\mathrm{that},\:\mathrm{you}\:\mathrm{thinked}\:\frac{{d}}{{dz}}{e}^{{z}} \:\mathrm{as}\:\mathrm{an}\:\mathrm{Algebra}\:\mathrm{question}. \\ $$$$\mathrm{Please},\:\mathrm{let}\:\mathrm{that}\:\mathrm{notation}\:\mathrm{go}\:\mathrm{to}\:\mathrm{your}\:\mathrm{mind}. \\ $$

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