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d-dx-e-tan-x-Z-A-




Question Number 164366 by Zaynal last updated on 16/Jan/22
(d/dx) (e^(tan(x)) )  {Z.A}
$$\frac{\boldsymbol{{d}}}{\boldsymbol{{dx}}}\:\left(\boldsymbol{{e}}^{\boldsymbol{{tan}}\left(\boldsymbol{{x}}\right)} \right) \\ $$$$\left\{\boldsymbol{{Z}}.\boldsymbol{\mathrm{A}}\right\} \\ $$
Commented by cortano1 last updated on 16/Jan/22
 y = e^(tan x)    ln y = tan x   ((y′)/y) = sec^2 x   (dy/dx) = sec^2 x (e^(tan x) )
$$\:{y}\:=\:{e}^{\mathrm{tan}\:{x}} \\ $$$$\:\mathrm{ln}\:{y}\:=\:\mathrm{tan}\:{x} \\ $$$$\:\frac{{y}'}{{y}}\:=\:\mathrm{sec}\:^{\mathrm{2}} {x} \\ $$$$\:\frac{{dy}}{{dx}}\:=\:\mathrm{sec}\:^{\mathrm{2}} {x}\:\left({e}^{\mathrm{tan}\:{x}} \right)\: \\ $$
Answered by muneer0o0 last updated on 16/Jan/22
(d/dx) (e^(tan(x)) )=   ((d(e^(tan x) ))/(d(tan x)))×((d(tan x))/dx) = e^(tan x)   sec^2 x
$$\frac{\boldsymbol{{d}}}{\boldsymbol{{dx}}}\:\left(\boldsymbol{{e}}^{\boldsymbol{{tan}}\left(\boldsymbol{{x}}\right)} \right)=\:\:\:\frac{{d}\left({e}^{\mathrm{tan}\:{x}} \right)}{{d}\left(\mathrm{tan}\:{x}\right)}×\frac{{d}\left(\mathrm{tan}\:{x}\right)}{{dx}}\:=\:{e}^{\mathrm{tan}\:{x}} \:\:\mathrm{sec}^{\mathrm{2}} {x}\: \\ $$

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