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Category: Differentiation

Show-that-the-curve-y-ln-5-7x-8-x-has-no-stationary-point-for-all-real-values-of-x-

Question Number 8244 by lepan last updated on 04/Oct/16 $${Show}\:{that}\:{the}\:{curve}\:{y}={ln}\left(\frac{\mathrm{5}−\mathrm{7}{x}}{\mathrm{8}+{x}}\right)\:{has} \\ $$$${no}\:{stationary}\:{point}\:{for}\:{all}\:{real}\:{values} \\ $$$${of}\:{x}. \\ $$ Answered by 123456 last updated on 06/Oct/16 $${y}=\mathrm{ln}\:\frac{\mathrm{5}−\mathrm{7}{x}}{\mathrm{8}+{x}} \\…

show-that-for-all-integer-n-n-1-divides-2n-n-

Question Number 73755 by ~blr237~ last updated on 15/Nov/19 $${show}\:{that}\:\:\:{for}\:{all}\:{integer}\:\:{n}\:,\:\:{n}+\mathrm{1}\:{divides}\:\begin{pmatrix}{\mathrm{2}{n}}\\{{n}}\end{pmatrix} \\ $$ Answered by mind is power last updated on 15/Nov/19 $$\begin{pmatrix}{\mathrm{2}{n}}\\{{n}}\end{pmatrix}=\frac{\left(\mathrm{2}{n}\right)!}{{n}!{n}!}\in\mathbb{N} \\ $$$$\frac{{n}}{{n}+\mathrm{1}}\begin{pmatrix}{\mathrm{2}{n}}\\{{n}}\end{pmatrix}=\frac{\mathrm{2}{n}!}{{n}!.\left({n}\right)!}.\frac{{n}}{{n}+\mathrm{1}}=\frac{\mathrm{2}{n}!}{\left({n}−\mathrm{1}\right)!.\left({n}+\mathrm{1}\right)!}=\begin{pmatrix}{\mathrm{2}{n}}\\{{n}+\mathrm{1}}\end{pmatrix} \\…

Question-139171

Question Number 139171 by mnjuly1970 last updated on 23/Apr/21 Answered by Dwaipayan Shikari last updated on 23/Apr/21 $$\Gamma\left(\mathrm{1}+{x}\right)=\mathrm{1}−\gamma{x}+{O}\left({x}^{\mathrm{2}} \right)\Rightarrow\Gamma\left({x}\right)=\frac{\mathrm{1}}{{x}}−\gamma+{O}\left({x}\right) \\ $$$$\Gamma'\left(\mathrm{1}+{x}\right)=−\gamma+{O}\left({x}\right)\Rightarrow{x}\Gamma\left({x}\right)\psi\left({x}+\mathrm{1}\right)=−\gamma+{O}\left({x}\right) \\ $$$$\Rightarrow{x}\Gamma\left({x}\right)\psi\left({x}\right)+\Gamma\left({x}\right)=−\gamma+{O}\left({x}\right)\Rightarrow\psi\left({x}\right)\sim−\frac{\gamma}{{x}\Gamma\left({x}\right)}−\frac{\mathrm{1}}{{x}} \\ $$$$\underset{{x}\rightarrow\mathrm{0}}…

let-f-x-tanx-tan2x-Find-the-points-of-discontinuity-of-f-on-0-2pi-and-determine-wether-each-duscontinuity-is-a-point-discontinuity-a-jump-discontinuity-or-a-vertical-asymtote-

Question Number 73569 by Rio Michael last updated on 13/Nov/19 $${let}\:{f}\left({x}\right)\:=\:\frac{{tanx}}{{tan}\mathrm{2}{x}}\:.\:{Find}\:{the}\:{points}\:{of}\:{discontinuity} \\ $$$${of}\:{f}\:{on}\:\left[\mathrm{0},\mathrm{2}\pi\right]\:{and}\:{determine}\:{wether}\:{each}\:{duscontinuity}\:{is} \\ $$$${a}\:{point}\:{discontinuity},{a}\:{jump}\:{discontinuity},{or}\:{a}\:{vertical}\:{asymtote} \\ $$$$ \\ $$ Commented by Rio Michael last updated…

prove-that-1-itan-i-cot-itan-pleas-sir-help-me-

Question Number 73560 by mhmd last updated on 13/Nov/19 $${prove}\:{that}\:\left(\mathrm{1}−{itan}\theta\right)/\left({i}+{cot}\theta\right)={itan}\theta \\ $$$${pleas}\:{sir}\:{help}\:{me}? \\ $$ Commented by MJS last updated on 13/Nov/19 $$\mathrm{it}'\mathrm{s}\:\mathrm{not}\:\mathrm{generally}\:\mathrm{true} \\ $$$$\frac{\mathrm{1}−\mathrm{i}\:\mathrm{tan}\:\theta}{\mathrm{i}+\mathrm{cot}\:\theta}=\mathrm{2sin}\:\theta\:\mathrm{cos}\:\theta\:−\mathrm{tan}\:\theta\:−\mathrm{2i}\:\mathrm{sin}^{\mathrm{2}} \:\theta…