Question Number 82245 by jagoll last updated on 19/Feb/20 $${find}\:{the}\:{solution}\: \\ $$$${x}\:\mathrm{sin}\:\left(\frac{{y}}{{x}}\right)\:{dy}\:=\:\left[{y}\:\mathrm{sin}\:\left(\frac{{y}}{{x}}\right)\:−{x}\right]\:{dx} \\ $$ Commented by john santu last updated on 19/Feb/20 $${let}\:{v}\:=\:\frac{{y}}{{x}}\:\Rightarrow\:{y}\:=\:{vx} \\ $$$$\frac{{dy}}{{dx}}\:=\:{v}\:+\:\frac{{dv}}{{dx}}\:\Rightarrow\:{dy}\:={v}\:{dx}+\:{dv}\:…
Question Number 16689 by Nayon last updated on 25/Jun/17 $${why}\:{any}\:{infinitely}\:{differentiable}\:{function}\:{is}\:{a}\:{power}\:{series}? \\ $$ Answered by prakash jain last updated on 25/Jun/17 $$\mathrm{Taylor}\:\mathrm{Series}\:\mathrm{for}\:\mathrm{class}\:\mathrm{C}^{\infty} . \\ $$ Terms…
Question Number 147753 by Odhiambojr last updated on 23/Jul/21 $${Find}\:{a}\:{point}\:{on}\:{the}\:{curve} \\ $$$$\:{y}={x}^{\mathrm{2}} −\mathrm{2}{x}−\mathrm{3}\:{at}\:{which}\:{the}\:{tangent}\:{is} \\ $$$$\:{parallel}\:{to}\:{the}\:{x}-{axis} \\ $$ Answered by Olaf_Thorendsen last updated on 23/Jul/21 $$\frac{{dy}}{{dx}}\:=\:\mathrm{2}{x}−\mathrm{2}\:=\:\mathrm{2}\left({x}−\mathrm{1}\right)…
Question Number 147748 by Odhiambojr last updated on 23/Jul/21 $${prove}\:{that}\:{curves}\:{x}^{\mathrm{2}\:} −{y}^{\mathrm{2}} =\mathrm{3}\:{and} \\ $$$$\:{xy}=\mathrm{2}\:{intersect}\:{at}\:{the}\:{right}\:{angle}\: \\ $$$$ \\ $$ Commented by Olaf_Thorendsen last updated on 23/Jul/21…
Question Number 147713 by Odhiambojr last updated on 22/Jul/21 $${Find}\:{a}\:{point}\:{on}\:{the}\:{curve}\:{y}=\sqrt{{x}} \\ $$$${where}\:{the}\:{tangent}\:{makes}\:{an}\:{angle}\: \\ $$$$\mathrm{45}°\:{with}\:{the}\:{positive}\:{x}-{axis} \\ $$ Answered by Olaf_Thorendsen last updated on 22/Jul/21 $$\frac{{dy}}{{dx}}\:=\:\mathrm{tan}\left(\theta\left({x}\right)\right)\:=\:\frac{\mathrm{1}}{\mathrm{2}\sqrt{{x}}} \\…
Question Number 82109 by jagoll last updated on 18/Feb/20 $${what}\:{is}\:{solution} \\ $$$$\frac{{dy}}{{dx}}\:=\:\mathrm{sin}\:\left({x}+{y}\right) \\ $$ Commented by mr W last updated on 18/Feb/20 $${u}={x}+{y} \\ $$$${y}={u}−{x}…
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Question Number 82059 by jagoll last updated on 18/Feb/20 $${what}\:{is}\:{derivative}\:{of}\:\:{h}\:=\:\sqrt{{ln}\left({x}\right)} \\ $$$${by}\:{first}\:{principle}\:{method}\: \\ $$ Answered by Henri Boucatchou last updated on 18/Feb/20 $$\frac{\mathrm{dh}\left(\mathrm{x}\right)}{\mathrm{dx}}=\frac{\mathrm{d}\sqrt{\mathrm{lnx}}}{\mathrm{dx}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:=\frac{\frac{\mathrm{dlnx}}{\mathrm{dx}}}{\mathrm{2}\sqrt{\mathrm{lnx}}}…
Question Number 82018 by TawaTawa last updated on 17/Feb/20 $$\mathrm{Differentiate}\:\:\:\:\:\mathrm{y}\:\:=\:\:\mathrm{2}^{\mathrm{x}} \:\:\:\:\mathrm{from}\:\mathrm{the}\:\mathrm{first}\:\mathrm{principle}. \\ $$ Commented by john santu last updated on 17/Feb/20 $${ln}\:{y}\:=\:{x}\:{ln}\:\mathrm{2} \\ $$$$\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\:{ln}\left({y}\right)\:\:=\:\underset{{h}\rightarrow\mathrm{0}}…
Question Number 81954 by Cmr 237 last updated on 16/Feb/20 $$\left.\:{soit}\:\alpha\in\right]\mathrm{0};\pi\left[.\:{determiner}:\right. \\ $$$$\left.\mathrm{1}\right){le}\:{module}\:{et}\:{l}'{argument}\:{de}: \\ $$$$\left.\boldsymbol{{a}}\left.\right)\mathrm{1}−\boldsymbol{{e}}^{\boldsymbol{{i}}\alpha} ,\boldsymbol{{b}}\right)\mathrm{1}+\boldsymbol{{e}}^{\boldsymbol{{i}\alpha}} \\ $$$$\left.\mathrm{2}\right)\boldsymbol{{deduire}}\:\boldsymbol{{le}}\:\boldsymbol{{module}}\:\boldsymbol{{et}}\:\boldsymbol{{l}}'\boldsymbol{{argument}}\:\boldsymbol{{de}} \\ $$$$\left.\:\left.\boldsymbol{{a}}\right)\:\frac{\mathrm{1}−\boldsymbol{{e}}^{\boldsymbol{{i}}\alpha} }{\mathrm{1}+{e}^{{i}\alpha} },\:{b}\right)\left(\mathrm{1}−{e}^{{i}\alpha} \right)\left(\mathrm{1}+{e}^{{i}\alpha} \right) \\…