Question Number 5171 by FilupSmith last updated on 25/Apr/16 $$\mathrm{Can}\:\mathrm{you}\:\mathrm{show}\:\mathrm{why}: \\ $$$$\left({a}\right)\:\:\:\:\:\frac{{d}}{{dx}}\left(\mathrm{sin}\:{x}\right)=\mathrm{cos}\:{x} \\ $$$$\left({b}\right)\:\:\:\:\:\frac{{d}}{{dx}}\left(\mathrm{cos}\:{x}\right)=−\mathrm{sin}\:{x} \\ $$ Commented by FilupSmith last updated on 25/Apr/16 $${f}\:'\left({x}\right)=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{f}\left({x}+{h}\right)−{f}\left({x}\right)}{{h}}…
Question Number 70705 by sadimuhmud 136 last updated on 07/Oct/19 $$\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{log}_{\mathrm{10}} \mathrm{x}\right)=? \\ $$ Answered by $@ty@m123 last updated on 07/Oct/19 $$\frac{\mathrm{d}}{\mathrm{dx}}\left(\frac{\mathrm{log}_{{e}} \:{x}.}{\mathrm{log}_{{e}} \:\mathrm{10}}\right)=\frac{\mathrm{1}}{\mathrm{ln}\:\mathrm{10}}.\frac{{d}}{{dx}}\left(\mathrm{ln}\:{x}\right) \\…
Question Number 136225 by mnjuly1970 last updated on 19/Mar/21 $$\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:…..{nice}\:\:\:{calculus}\:\: \\ $$$$\:\:\:\: \\ $$$$\:\:\:\:\:{find}\:::\:\Omega=\int_{\mathrm{0}} ^{\:\infty} {xe}^{−{x}} {coth}\left(\frac{{x}}{\mathrm{2}}\right){dx}=? \\ $$$$ \\ $$$$ \\ $$$$…
Question Number 70601 by oyemi kemewari last updated on 06/Oct/19 $$\mathrm{prove}\:\mathrm{thst}\:\mathrm{I}=\int_{\mathrm{0}} ^{\infty} \mathrm{f}\left(\mathrm{x}+\frac{\mathrm{1}}{×}\right)\frac{\mathrm{lnx}}{\mathrm{x}}\mathrm{dx}=\mathrm{0} \\ $$ Answered by mind is power last updated on 06/Oct/19 $${let}\:{x}=\frac{\mathrm{1}}{{y}}\Rightarrow{I}=\int_{\infty}…
Question Number 136138 by bramlexs22 last updated on 19/Mar/21 $${What}\:{is}\:{max}\:{and}\:{min}\:{value} \\ $$$${of}\:{y}=\mathrm{4}−{x}^{\mathrm{2}} −\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }\:? \\ $$ Answered by ajfour last updated on 19/Mar/21 $${let}\:\:\mathrm{1}−{x}^{\mathrm{2}} ={s}\geqslant\mathrm{0}…
Question Number 70582 by Raphael last updated on 05/Oct/19 $$\mathrm{Given}\:\mathrm{that}\:\mathrm{y}=\frac{\mathrm{4}}{\:\sqrt{\left(\mathrm{x}^{\mathrm{3}} +\mathrm{1}\right)}}\:\mathrm{show}\:\mathrm{that}\:\mathrm{2}\left(\mathrm{x}^{\mathrm{3}} +\mathrm{1}\right)\frac{\mathrm{dy}}{\mathrm{dx}}=−\mathrm{3x}^{\mathrm{2}} \mathrm{y}. \\ $$ Commented by kaivan.ahmadi last updated on 05/Oct/19 $$\mathrm{2}\left({x}^{\mathrm{3}} +\mathrm{1}\right)×\frac{−\frac{\mathrm{4}×\mathrm{3}{x}^{\mathrm{2}} }{\mathrm{2}\sqrt{{x}^{\mathrm{3}}…
Question Number 70583 by Raphael last updated on 05/Oct/19 $$\mathrm{Given}\:\mathrm{that}\:\mathrm{y}=\frac{\mathrm{4}}{\:\sqrt{\left(\mathrm{x}^{\mathrm{3}} +\mathrm{1}\right)}}\:\mathrm{show}\:\mathrm{that}\:\mathrm{2}\left(\mathrm{x}^{\mathrm{3}} +\mathrm{1}\right)\frac{\mathrm{dy}}{\mathrm{dx}}=−\mathrm{3x}^{\mathrm{2}} \mathrm{y}. \\ $$ Answered by MJS last updated on 05/Oct/19 $${dy}=−\frac{\mathrm{6}{x}^{\mathrm{2}} }{\left({x}^{\mathrm{3}} +\mathrm{1}\right)^{\mathrm{3}/\mathrm{2}}…
Question Number 4993 by love math last updated on 29/Mar/16 $${f}\left({x}\right)=\frac{{x}\:{ln}\:{x}}{\mathrm{1}+\:{e}^{{x}} } \\ $$$$ \\ $$$${f}\:'\left({x}\right)\:−\:? \\ $$ Answered by FilupSmith last updated on 30/Mar/16…
Question Number 4992 by love math last updated on 29/Mar/16 $${f}\left({x}\right)=\mathrm{2}{x}\:{e}^{{x}} \\ $$$${f}\:'\left({x}\right)\:−? \\ $$ Answered by FilupSmith last updated on 29/Mar/16 $${f}\left({x}\right)={uv} \\ $$$${f}\left({x}\right)={uv}'+{u}'{v}…
Question Number 136006 by liberty last updated on 17/Mar/21 $$ \\ $$What are the dimensions of the rectangle of maximum area that can be inscribed…