Question Number 45993 by peter frank last updated on 19/Oct/18 Commented by math khazana by abdo last updated on 20/Oct/18 $$\frac{{d}}{{dx}}\left({e}^{{x}^{\mathrm{2}} } \right)={lim}_{{h}\rightarrow\mathrm{0}} \:\frac{{e}^{\left({x}+{h}\right)^{\mathrm{2}} }…
Question Number 176809 by cortano1 last updated on 27/Sep/22 $$\:\:\begin{cases}{\mathrm{x}=\mathrm{cos}\:^{\mathrm{3}} \emptyset}\\{\mathrm{y}=\mathrm{sin}\:^{\mathrm{3}} \emptyset}\end{cases}\:\Rightarrow\frac{\mathrm{d}^{\mathrm{2}} \mathrm{y}}{\mathrm{dx}^{\mathrm{2}} }\:=? \\ $$ Answered by mr W last updated on 27/Sep/22 $$\frac{{dy}}{{d}\phi}=−\mathrm{3}\:\mathrm{cos}^{\mathrm{2}}…
Question Number 111213 by mnjuly1970 last updated on 02/Sep/20 Answered by mathmax by abdo last updated on 02/Sep/20 $$\mathrm{let}\:\mathrm{S}_{\mathrm{n}} =\sum_{\mathrm{k}=\mathrm{1}} ^{\mathrm{2n}} \:\mathrm{cos}^{\mathrm{2}} \left(\frac{\mathrm{k}\pi}{\mathrm{2n}+\mathrm{1}}\right)\:\Rightarrow\mathrm{S}_{\mathrm{n}} =\frac{\mathrm{1}}{\mathrm{2}}\sum_{\mathrm{k}=\mathrm{1}} ^{\mathrm{2n}}…
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Question Number 45464 by peter frank last updated on 13/Oct/18 $$\boldsymbol{\mathrm{D}}\mathrm{ifferentiate}\:\mathrm{with}\:\mathrm{respect}\:\mathrm{to}\:\boldsymbol{\mathrm{x}} \\ $$$$\boldsymbol{\mathrm{arctan}}\left(\frac{\boldsymbol{\mathrm{a}}^{\mathrm{2}} +\boldsymbol{\mathrm{x}}^{\mathrm{2}} }{\boldsymbol{\mathrm{a}}^{\mathrm{2}} −\boldsymbol{\mathrm{x}}^{\mathrm{2}} }\right) \\ $$ Commented by peter frank last updated…
Question Number 45456 by peter frank last updated on 13/Oct/18 Commented by peter frank last updated on 13/Oct/18 $$\mathrm{math1967} \\ $$ Commented by math1967 last…
Question Number 45281 by peter frank last updated on 11/Oct/18 Commented by math1967 last updated on 11/Oct/18 $${pls}\:{chq}.\:{q}.{no}\mathrm{1} \\ $$ Commented by peter frank last…
Question Number 45280 by peter frank last updated on 11/Oct/18 $$\boldsymbol{\mathrm{solve}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{simultaneous}}\:\boldsymbol{\mathrm{equation}} \\ $$$$\left.\boldsymbol{\mathrm{a}}\right)\boldsymbol{\mathrm{sin}}\left(\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{y}}\right)=\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}\:\:\:} \\ $$$$\boldsymbol{\mathrm{cos}}\mathrm{2}\boldsymbol{\mathrm{x}}=\frac{-\mathrm{1}\:\:}{\mathrm{2}}\:\boldsymbol{\mathrm{for}}\:\boldsymbol{\mathrm{x}}\:\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{y}}\:\boldsymbol{\mathrm{ranging}}\:\boldsymbol{\mathrm{from}}\:\mathrm{0}\:\boldsymbol{\mathrm{to}}\:\mathrm{360}\:\boldsymbol{\mathrm{inclusive}} \\ $$$$\left.\boldsymbol{\mathrm{b}}\right)\boldsymbol{\mathrm{if}}\:\boldsymbol{\mathrm{siny}}+\boldsymbol{\mathrm{cosx}}=\boldsymbol{\mathrm{x}}\:\:\boldsymbol{\mathrm{show}}\:\boldsymbol{\mathrm{that}}\:\frac{\boldsymbol{\mathrm{d}}^{\mathrm{2}} \boldsymbol{\mathrm{y}}}{\boldsymbol{\mathrm{dx}}^{\mathrm{2}\:} }\:=\frac{\boldsymbol{\mathrm{x}}}{\left(\mathrm{2}−\boldsymbol{\mathrm{x}}^{\mathrm{2}} \right)^{\frac{\mathrm{3}}{\mathrm{2}}} } \\ $$ Answered by…
Question Number 176274 by mnjuly1970 last updated on 15/Sep/22 Answered by Frix last updated on 16/Sep/22 $${y}=\mathrm{coth}\:{x}\:\Leftrightarrow\:{x}=\frac{\mathrm{1}}{\mathrm{2}}\mathrm{ln}\:\frac{{y}+\mathrm{1}}{{y}−\mathrm{1}} \\ $$$$\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}\mathrm{coth}^{−\mathrm{1}} \:\sqrt{\mathrm{2}}\:=\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\mathrm{ln}\:\left(\mathrm{1}+\sqrt{\mathrm{2}}\right) \\ $$$$\underset{\mathrm{0}} {\overset{\pi} {\int}}\frac{\mathrm{sin}^{\mathrm{2}} \:{x}}{\:\sqrt{\mathrm{tan}^{\mathrm{2}}…