Question Number 142681 by mnjuly1970 last updated on 03/Jun/21 $$ \\ $$$$\:\:\:\:{prove}\:\:{that}\::: \\ $$$$\:\:\:\:\:\:\boldsymbol{\phi}:=\int_{\mathrm{0}} ^{\:\infty} \frac{{ln}\left(\mathrm{1}+{cos}\left({x}\right)\right)}{\mathrm{1}+{e}^{\:{x}} }{dx}=\mathrm{0} \\ $$$$\:\:\:\:\:……………. \\ $$$$ \\ $$ Commented by…
Question Number 11581 by Nayon last updated on 28/Mar/17 $${cAn}\:{you}\:{prove}\:{the}\:{chAin}\:{role}?? \\ $$ Answered by linkelly0615 last updated on 28/Mar/17 $${yes}\sim \\ $$ Commented by linkelly0615…
Question Number 142655 by mnjuly1970 last updated on 03/Jun/21 $$\:\:{evaluate}….. \\ $$$$\:\:\:\:\:\:\:\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left(\frac{{n}.{cos}\left({n}\pi\right)}{\Gamma\:\left(\mathrm{2}{n}+\mathrm{2}\right)}\right)=?\:….. \\ $$$$\:\:\:……. \\ $$ Answered by qaz last updated on 03/Jun/21…
Question Number 11580 by Nayon last updated on 28/Mar/17 $${why}\:\frac{{dy}}{{dx}}=\frac{{dy}}{{du}}.\frac{{du}}{{dx}} \\ $$ Answered by mrW1 last updated on 28/Mar/17 $${the}\:{chain}\:{rule}\:{is}\:{one}\:{of}\:{the}\:{elementary} \\ $$$${rules}.\:{you}\:{should}\:{know}\:{them},\:{but}\:\: \\ $$$${you}\:{don}'{t}\:{need}\:{to}\:{prove}\:{them}.\:{if} \\…
Question Number 11571 by Nayon last updated on 28/Mar/17 $${why}\:\:\:\frac{{d}\left[{f}\left\{{g}\left({x}\right)\right\}\right]}{{dx}}=\frac{{df}\left[\left\{{g}\left({x}\right)\right\}\right]}{{dg}\left({x}\right)}.\frac{{dg}\left({x}\right)}{{dx}}? \\ $$ Answered by mrW1 last updated on 28/Mar/17 $${y}={f}\left({u}\right) \\ $$$${u}={g}\left({x}\right) \\ $$$$\frac{{dy}}{{dx}}=\frac{{dy}}{{du}}×\frac{{du}}{{dx}} \\…
Question Number 11567 by Nayon last updated on 28/Mar/17 $${why}\:\:\:\:\:\:{li}\underset{{x}\rightarrow\mathrm{0}} {{m}}\:\frac{{f}\left({x}\right)}{{g}\left({x}\right)}={li}\underset{{x}\rightarrow\mathrm{0}} {{m}}\frac{{f}'\left({x}\right)}{{g}'\left({x}\right)} \\ $$ Answered by mrW1 last updated on 28/Mar/17 $${l}'{hopital}'{s}\:{rule} \\ $$ Terms…
Question Number 11565 by Nayon last updated on 28/Mar/17 $${if}\:\frac{{dy}}{{dx}}={p}\:\:,{then}\:{why}\:{dy}={pdx}? \\ $$ Answered by mrW1 last updated on 28/Mar/17 $${if}\:\frac{{dy}}{{dx}}={p}\Rightarrow{y}={px} \\ $$$$\Rightarrow{dy}={pdx} \\ $$ Terms…
Question Number 11563 by Nayon last updated on 28/Mar/17 $${if}\:\:{f}\left({x}\right)={g}\left({y}\right) \\ $$$${then}\:{why}\:\frac{{d}}{{dx}}\left({f}\left({x}\right)\right)=\frac{{d}}{{dy}}\left({g}\left({y}\right)\right)? \\ $$ Commented by mrW1 last updated on 28/Mar/17 $${this}\:{is}\:{not}\:{always}\:{correct}.\:{but} \\ $$$$\frac{{d}}{{dx}}\left({f}\left({x}\right)\right)=\frac{{d}}{{dy}}\left({g}\left({y}\right)\right)×\frac{{dy}}{{dx}} \\…
Question Number 11552 by Nayon last updated on 28/Mar/17 $${find}\:\frac{{d}}{{dx}}\left({y}\right)\:{where}\:\:{y}=\overset{\sqrt{{x}}} {\:}\sqrt{\sqrt{{x}}} \\ $$$$ \\ $$ Answered by ajfour last updated on 28/Mar/17 $${y}\:=\:\left(\sqrt{{x}}\right)^{\frac{\mathrm{1}}{\:\sqrt{{x}}}} \\ $$$$\mathrm{ln}\:{y}\:=\frac{\mathrm{ln}\:\sqrt{{x}}}{\:\sqrt{{x}}}…
Question Number 11547 by Nayon last updated on 28/Mar/17 $$\:\:\:\:\:\:\:\:\:\:\frac{{d}}{{dx}}\left({x}^{{x}} \right)=?\left[{please}\:{give}\:{the}\:{answer}\:{with}\:{proof}\right] \\ $$ Answered by sma3l2996 last updated on 28/Mar/17 $${x}^{{x}} ={e}^{{xln}\left({x}\right)} \\ $$$${so}\:\:\frac{{d}\left({x}^{{x}} \right)}{{dx}}=\frac{{d}\left({e}^{{xln}\left({x}\right)}…