Question Number 13068 by tawa tawa last updated on 13/May/17 $$\mathrm{If}\:\:\:\mathrm{y}\:=\:\:\sqrt[{\mathrm{3}}]{\mathrm{x}}\:\:\:\:\:\mathrm{Find}\:\:\frac{\mathrm{dy}}{\mathrm{dx}}\:\:\mathrm{from}\:\mathrm{the}\:\mathrm{first}\:\mathrm{principle} \\ $$ Answered by ajfour last updated on 13/May/17 $$\frac{{dy}}{{dx}}=\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\left({x}+{h}\right)^{\mathrm{1}/\mathrm{3}} −{x}^{\mathrm{1}/\mathrm{3}} }{{h}} \\…
Question Number 78581 by john santu last updated on 18/Jan/20 $$\int\:\frac{\mathrm{2}{x}\mathrm{sin}\:\mathrm{2}{x}}{\left(\mathrm{2}{x}−\mathrm{sin}\:\mathrm{2}{x}\right)^{\mathrm{2}} }\:{dx}\:? \\ $$ Commented by john santu last updated on 19/Jan/20 $${consider}\: \\ $$$$\frac{\mathrm{2}{x}\:\mathrm{sin}\:\mathrm{2}{x}}{\left(\mathrm{2}{x}−\mathrm{sin}\:\mathrm{2}{x}\right)^{\mathrm{2}}…
Question Number 424 by 123456 last updated on 25/Jan/15 $${f}\left({x},{y}\right)=\frac{\left({x}+{y}\right)^{\mathrm{2}} }{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} } \\ $$$$\begin{cases}{\frac{\partial{f}}{\partial{u}}=?}\\{\frac{\partial{f}}{\partial{v}}=?}\end{cases} \\ $$$$\begin{cases}{{x}={uv}}\\{{y}={u}+{v}}\end{cases} \\ $$ Answered by prakash jain last updated…
Question Number 131481 by bemath last updated on 05/Feb/21 $$\mathrm{In}\:\mathrm{1790}\:\mathrm{the}\:\mathrm{population}\:\mathrm{of}\:\mathrm{the}\: \\ $$$$\mathrm{United}\:\mathrm{States}\:\mathrm{was}\:\mathrm{3}.\mathrm{93}\:\mathrm{million} \\ $$$$\mathrm{and}\:\mathrm{in}\:\mathrm{1890}\:\mathrm{it}\:\mathrm{was}\:\mathrm{62}.\mathrm{98}\:\mathrm{million} \\ $$$$\mathrm{Using}\:\mathrm{the}\:\mathrm{Malthusian}\:\mathrm{model}\:, \\ $$$$\mathrm{estimate}\:\mathrm{the}\:\mathrm{U}.\mathrm{S}\:\mathrm{population}\:\mathrm{as} \\ $$$$\mathrm{a}\:\mathrm{function}\:\mathrm{of}\:\mathrm{time} \\ $$ Answered by EDWIN88…
Question Number 131470 by bemath last updated on 05/Feb/21 $${Consider}\:{a}\:{large}\:{tank}\:{holding} \\ $$$$\mathrm{1000}\:{L}\:{of}\:{pure}\:{water}\:{into} \\ $$$${which}\:{a}\:{brine}\:{solution}\:{of} \\ $$$${salt}\:{begins}\:{to}\:{flow}\:{at}\:{constant} \\ $$$${rate}\:{of}\:\mathrm{6}{L}/{min}.\:{The}\:{solution} \\ $$$${inside}\:{the}\:{tank}\:{is}\:{kept}\:{well} \\ $$$${stirred}\:{and}\:{is}\:{flowing}\:{out}\:{of} \\ $$$${the}\:{tank}\:{at}\:{rate}\:{of}\:\mathrm{6}{L}/{min}. \\…
Question Number 352 by Vishal Bhardwaj last updated on 25/Jan/15 $${Q}.\:\:{Define}\:\bullet\bullet\:\:{LHD}\left({left}\:{hand}\right. \\ $$$$\left.{derivative}\right)\:=\:{f}\left({a}−\mathrm{0}\right) \\ $$$$\underset{{h}\rightarrow\mathrm{0}} {{lim}}\:\frac{{f}\left({a}−{h}\right)−{f}\left({a}\right)}{−{h}} \\ $$$$ \\ $$$$ \\ $$ Commented by prakash…
Question Number 356 by 123456 last updated on 25/Jan/15 $${f}_{\mathrm{0}} \left({x}\right)=\mathrm{1} \\ $$$${f}_{\mathrm{1}} \left({x}\right)={x} \\ $$$${f}_{{n}+\mathrm{1}} \left({x}\right)={x}^{{f}_{{n}} \left({x}\right)} \\ $$$$\frac{\partial{f}_{{n}} }{\partial{x}}=?,{n}\in\mathbb{N}^{\ast} \\ $$ Answered by…
Question Number 295 by defg last updated on 25/Jan/15 $$\mathrm{If}\:{V}=\mathrm{log}\:\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right)\:\mathrm{then}\:{V}_{{xx}} +{V}_{{yy}} =? \\ $$ Answered by 123456 last updated on 19/Dec/14 $${V}_{{x}} =\frac{\mathrm{2}{x}}{{x}^{\mathrm{2}}…
Question Number 281 by samarth last updated on 25/Jan/15 $$\mathrm{If}\:{f}\left({x}\right)=\mathrm{tan}^{−\mathrm{1}} \sqrt{\frac{\mathrm{1}+\mathrm{sin}\:{x}}{\mathrm{1}−\mathrm{sin}\:{x}}},\:\mathrm{0}\leqslant{x}\leqslant\pi/\mathrm{2},\:\mathrm{then}\:{f}\:'\left(\pi/\mathrm{6}\right)=? \\ $$ Answered by 123456 last updated on 18/Dec/14 $${f}\left({x}\right)=\mathrm{tan}^{−\mathrm{1}} \sqrt{\frac{\mathrm{1}+\mathrm{sin}\:{x}}{\mathrm{1}−\mathrm{sin}\:{x}}} \\ $$$$\frac{\partial{f}}{\partial{x}}=\frac{\partial}{\partial{x}}\left(\mathrm{tan}^{−\mathrm{1}} \sqrt{\frac{\mathrm{1}+\mathrm{sin}\:{x}}{\mathrm{1}−\mathrm{sin}\:{x}}}\right)…
Question Number 131275 by Engr_Jidda last updated on 03/Feb/21 $${Orthogonal}\:{and}\:{Orthonormal}\:{sets} \\ $$$${given}\:{that}\:\:{y}_{{n}} \left({t}\right)=\varrho^{{t}} {B}_{{n}} {sin}\frac{{n}\pi}{\mathrm{4}}{t}\: \\ $$$${and}\:{y}_{{m}} \left({t}\right)=\varrho^{{t}} {B}_{{m}} {sin}\frac{{m}\pi}{\mathrm{4}}{t} \\ $$$${show}\:{that}\:\int_{\mathrm{0}} ^{\mathrm{4}} {y}_{{n}} \left({t}\right){y}_{{m}}…