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Category: Differentiation

nice-calculus-prove-that-lim-n-n-1-n-2-n-e-3-4-where-n-superfactorial-n-n-1-n-2-3-2-1-m-

Question Number 120312 by mnjuly1970 last updated on 30/Oct/20 $$\:\:\:\:\:\:\:\:\:\:…\:\clubsuit{nice}\:\:{calculus}\clubsuit… \\ $$$$\:\:\:\:\:\:{prove}\:\:{that}\::: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:{lim}_{{n}\rightarrow\infty} \frac{\sqrt[{{n}^{\mathrm{2}} }]{{n\$}}}{\:\sqrt{{n}}}?\overset{???} {=}{e}^{\frac{−\mathrm{3}}{\mathrm{4}}} \\ $$$$\:\:\:\:\:{where}\:::\:\:{n\$}\:\overset{{superfactorial}} {=}{n}!.\left({n}−\mathrm{1}\right)!.\left({n}−\mathrm{2}\right)!…\mathrm{3}!.\mathrm{2}!.\mathrm{1}! \\ $$$$\:\:\:\:\:\:\:\:…\spadesuit{m}.{n}.\mathrm{1970}\spadesuit… \\…

Question-120243

Question Number 120243 by benjo_mathlover last updated on 30/Oct/20 Answered by bemath last updated on 30/Oct/20 $${let}\:{f}\left({x}\right)\:=\:{x}^{\frac{\mathrm{1}}{{x}^{\mathrm{6}} }} \:\Leftrightarrow\:\mathrm{ln}\:{f}\left({x}\right)=\:\frac{\mathrm{ln}\:\left({x}\right)}{{x}^{\mathrm{6}} } \\ $$$${differentiating}\:{both}\:{sides}\:{gives} \\ $$$$\frac{{f}\:'\left({x}\right)}{{f}\left({x}\right)}\:=\:\frac{{x}^{\mathrm{5}} −\mathrm{6}{x}^{\mathrm{5}}…

Question-54679

Question Number 54679 by Raj Singh last updated on 09/Feb/19 Commented by maxmathsup by imad last updated on 09/Feb/19 $${let}\:{f}\left({x}\right)=\frac{\mathrm{1}}{\:\sqrt{{x}+{a}}}\:\:{for}\:{x}>−{a}\:\:{we}\:{have}\:{f}^{'} \left({x}\right)={lim}_{{h}\rightarrow\mathrm{0}} \:\frac{{f}\left({x}+{h}\right)−{f}\left({x}\right)}{{h}} \\ $$$$={lim}_{{h}\rightarrow\mathrm{0}} \:\:\frac{\frac{\mathrm{1}}{\:\sqrt{{x}+{h}+{a}}\:\:}−\frac{\mathrm{1}}{\:\sqrt{{x}+{a}}}}{{h}}\:={lim}_{{h}\rightarrow\mathrm{0}}…

Question-119969

Question Number 119969 by huotpat last updated on 28/Oct/20 Answered by Dwaipayan Shikari last updated on 28/Oct/20 $$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}{tan}^{−\mathrm{1}} \frac{\mathrm{1}}{{k}^{\mathrm{2}} +{k}+\mathrm{1}} \\ $$$$\underset{{k}=\mathrm{1}}…

i-1-2-cos-pi-7-1-2-cos-3pi-7-1-2-cos-9pi-7-ii-3-tan-1-3-tan-2-3-tan-3-3-tan-29-

Question Number 119937 by bobhans last updated on 28/Oct/20 $$\:\left({i}\right)\left(\frac{\mathrm{1}}{\mathrm{2}}−\mathrm{cos}\:\frac{\pi}{\mathrm{7}}\right)\left(\frac{\mathrm{1}}{\mathrm{2}}−\mathrm{cos}\:\frac{\mathrm{3}\pi}{\mathrm{7}}\right)\left(\frac{\mathrm{1}}{\mathrm{2}}−\mathrm{cos}\:\frac{\mathrm{9}\pi}{\mathrm{7}}\right)? \\ $$$$\left({ii}\right)\:\left(\sqrt{\mathrm{3}}+\mathrm{tan}\:\mathrm{1}°\right)\left(\sqrt{\mathrm{3}}+\mathrm{tan}\:\mathrm{2}°\right)\left(\sqrt{\mathrm{3}}+\mathrm{tan}\:\mathrm{3}°\right)×…×\left(\sqrt{\mathrm{3}}+\mathrm{tan}\:\mathrm{29}°\right)? \\ $$ Answered by TANMAY PANACEA last updated on 28/Oct/20 $$\left.{ii}\right)\:\sqrt{\mathrm{3}}\:+{tan}\mathrm{1}^{{o}} \\ $$$$={tan}\mathrm{60}^{{o}}…

Find-the-largest-possible-area-of-trapezoid-that-can-be-drawn-under-the-x-axis-so-that-one-of-its-bases-is-on-the-x-axis-and-the-other-two-vertices-are-on-the-curve-y-x-2-9-

Question Number 185453 by cortano1 last updated on 22/Jan/23 $$\:{Find}\:{the}\:{largest}\:{possible}\:{area} \\ $$$$\:{of}\:{trapezoid}\:{that}\:{can}\:{be}\:{drawn}\: \\ $$$$\:{under}\:{the}\:{x}−{axis}\:{so}\:{that}\:{one}\: \\ $$$$\:{of}\:{its}\:{bases}\:{is}\:{on}\:{the}\:{x}−{axis}\: \\ $$$$\:{and}\:{the}\:{other}\:{two}\:{vertices}\:{are} \\ $$$$\:{on}\:{the}\:{curve}\:{y}={x}^{\mathrm{2}} −\mathrm{9}\: \\ $$ Commented by…

Let-x-y-z-be-non-negative-real-numbers-such-that-x-y-z-1-Find-the-extremum-of-F-2x-2-y-3z-2-

Question Number 119724 by benjo_mathlover last updated on 26/Oct/20 $${Let}\:{x},{y},{z}\:{be}\:{non}\:{negative}\:{real}\:{numbers} \\ $$$${such}\:{that}\:{x}+{y}+{z}=\mathrm{1}.\:{Find}\:{the}\:{extremum} \\ $$$${of}\:{F}\:=\:\mathrm{2}{x}^{\mathrm{2}} +{y}+\mathrm{3}{z}^{\mathrm{2}} \:. \\ $$ Answered by mindispower last updated on 26/Oct/20…