Question Number 120463 by bramlexs22 last updated on 31/Oct/20 $${Find}\:{the}\:{slope}\:{of}\:{the}\:{line}\:{tangent} \\ $$$${to}\:{the}\:{graph}\:{r}=\mathrm{3cos}\:^{\mathrm{2}} \left(\mathrm{2}\theta\right)\:{at} \\ $$$$\theta=\frac{\pi}{\mathrm{6}}.\: \\ $$ Answered by mr W last updated on 31/Oct/20…
Question Number 120452 by john santu last updated on 31/Oct/20 $${Find}\:{the}\:{equation}\:{of}\:{the}\:{line}\:{tangent}\:{to}\:{the}\:{parametric} \\ $$$${curve}\:{given}\:{by}\:{the}\:{equations}\:\begin{cases}{{x}=\left(\mathrm{1}+{t}^{\mathrm{3}} \right)^{\mathrm{4}} +{t}^{\mathrm{2}} }\\{{y}={t}^{\mathrm{5}} +{t}^{\mathrm{2}} +\mathrm{2}}\end{cases}\:{at}\:{t}=\mathrm{1} \\ $$ Answered by physicstutes last updated…
Question Number 120378 by john santu last updated on 31/Oct/20 $${find}\:{the}\:{point}\:{on}\:{the}\:{curve}\:\mathrm{3}{x}^{\mathrm{2}} −\mathrm{4}{y}^{\mathrm{2}} =\mathrm{72} \\ $$$${closest}\:{to}\:{line}\:\mathrm{3}{x}+\mathrm{2}{y}−\mathrm{1}=\mathrm{0} \\ $$ Commented by bramlexs22 last updated on 31/Oct/20 $${let}\:\left({x},{y}\right)\:{is}\:{a}\:{point}\:{on}\:{hyperbola}…
Question Number 120322 by sdfg last updated on 30/Oct/20 Commented by TANMAY PANACEA last updated on 30/Oct/20 $${do}\:{x}'\:{means}\:=\frac{{dx}}{{dt}}\:{pls}… \\ $$ Commented by Dwaipayan Shikari last…
Question Number 120312 by mnjuly1970 last updated on 30/Oct/20 $$\:\:\:\:\:\:\:\:\:\:…\:\clubsuit{nice}\:\:{calculus}\clubsuit… \\ $$$$\:\:\:\:\:\:{prove}\:\:{that}\::: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:{lim}_{{n}\rightarrow\infty} \frac{\sqrt[{{n}^{\mathrm{2}} }]{{n\$}}}{\:\sqrt{{n}}}?\overset{???} {=}{e}^{\frac{−\mathrm{3}}{\mathrm{4}}} \\ $$$$\:\:\:\:\:{where}\:::\:\:{n\$}\:\overset{{superfactorial}} {=}{n}!.\left({n}−\mathrm{1}\right)!.\left({n}−\mathrm{2}\right)!…\mathrm{3}!.\mathrm{2}!.\mathrm{1}! \\ $$$$\:\:\:\:\:\:\:\:…\spadesuit{m}.{n}.\mathrm{1970}\spadesuit… \\…
Question Number 120243 by benjo_mathlover last updated on 30/Oct/20 Answered by bemath last updated on 30/Oct/20 $${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}} } \\ $$$${differentiating}\:{both}\:{sides}\:{gives} \\ $$$$\frac{{f}\:'\left({x}\right)}{{f}\left({x}\right)}\:=\:\frac{{x}^{\mathrm{5}} −\mathrm{6}{x}^{\mathrm{5}}…
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Question Number 54679 by Raj Singh last updated on 09/Feb/19 Commented by maxmathsup by imad last updated on 09/Feb/19 $${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}} \\ $$$$={lim}_{{h}\rightarrow\mathrm{0}} \:\:\frac{\frac{\mathrm{1}}{\:\sqrt{{x}+{h}+{a}}\:\:}−\frac{\mathrm{1}}{\:\sqrt{{x}+{a}}}}{{h}}\:={lim}_{{h}\rightarrow\mathrm{0}}…
Question Number 119969 by huotpat last updated on 28/Oct/20 Answered by Dwaipayan Shikari last updated on 28/Oct/20 $$\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}} \\ $$$$\underset{{k}=\mathrm{1}}…
Question Number 119965 by sdfg last updated on 28/Oct/20 Answered by Dwaipayan Shikari last updated on 28/Oct/20 $$\Gamma\left({x}\right)=\int_{\mathrm{0}} ^{\infty} {t}^{{x}−\mathrm{1}} {e}^{−{t}} {dt} \\ $$$$\frac{{d}\Gamma\left({x}\right)}{{dx}}=\int_{\mathrm{0}} ^{\infty}…