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Category: Differentiation

f-tan-x-2f-cot-x-4x-f-x-

Question Number 198562 by cortano12 last updated on 22/Oct/23 $$\:\:\:\mathrm{f}\left(\mathrm{tan}\:\mathrm{x}\right)+\:\mathrm{2f}\left(\mathrm{cot}\:\mathrm{x}\right)\:=\:\mathrm{4x}\: \\ $$$$\:\:\:\mathrm{f}\:'\left(\mathrm{x}\right)=\:? \\ $$ Answered by dimentri last updated on 22/Oct/23 $$\:{let}\:{x}=\frac{\pi}{\mathrm{2}}−{t}\: \\ $$$$\:{f}\left(\mathrm{cot}\:{t}\right)\:+\mathrm{2}\:{f}\left(\mathrm{tan}\:{t}\right)=\:\mathrm{2}\pi−\mathrm{4}{t} \\…

If-f-x-x-1-x-lnt-t-2-1-arctan-t-dt-Prove-that-x-gt-0-f-x-pi-8-pi-0-arctan-1-2-x-1-x-sint-dt-lim-x-f-x-pi-3-16-

Question Number 197384 by Erico last updated on 15/Sep/23 $$\mathrm{If}\:{f}\left({x}\right)=\underset{\:\frac{\mathrm{1}}{\mathrm{x}}} {\int}^{\:\:\mathrm{x}} \frac{{lnt}}{{t}^{\mathrm{2}} −\mathrm{1}}{arctan}\left({t}\right){dt} \\ $$$$\mathrm{Prove}\:\mathrm{that}: \\ $$$$\bullet\:\forall{x}>\mathrm{0}\:\:\:\:\:\:\:\:{f}\left({x}\right)=\:\frac{\pi}{\mathrm{8}}\:\underset{\:\mathrm{0}} {\int}^{\:\pi} \mathrm{arctan}\left[\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{x}−\frac{\mathrm{1}}{\mathrm{x}}\right)\mathrm{sint}\right]\mathrm{dt} \\ $$$$\bullet\underset{\mathrm{x}\rightarrow+\infty} {\mathrm{lim}}\:{f}\left({x}\right)=\frac{\pi^{\mathrm{3}} }{\mathrm{16}} \\ $$…

if-x-cos-u-y-sin-u-and-z-f-x-y-then-show-that-2-z-x-2-2-z-y-2-u-4-2-z-u-2-u-3-z-u-u-4-2-z-2-

Question Number 197317 by universe last updated on 13/Sep/23 $$\:\mathrm{if}\:\:\:\mathrm{x}\:\:=\:\:\frac{\mathrm{cos}\:\theta}{\mathrm{u}}\:\:,\:\mathrm{y}\:\:=\:\frac{\mathrm{sin}\:\theta}{\mathrm{u}}\:\:{and}\:\mathrm{z}\:\:=\:\:\mathrm{f}\left(\mathrm{x},\mathrm{y}\right) \\ $$$$\mathrm{then}\:\mathrm{show}\:\mathrm{that}\: \\ $$$$\:\:\:\frac{\partial^{\mathrm{2}} \mathrm{z}}{\partial\mathrm{x}^{\mathrm{2}} }\:+\:\frac{\partial^{\mathrm{2}} \mathrm{z}}{\partial\mathrm{y}^{\mathrm{2}} }\:=\:\mathrm{u}^{\mathrm{4}} \:\frac{\partial^{\mathrm{2}} \mathrm{z}}{\partial\mathrm{u}^{\mathrm{2}} }\:+\:\mathrm{u}^{\mathrm{3}} \:\frac{\partial\mathrm{z}}{\partial\mathrm{u}}\:+\:\mathrm{u}^{\mathrm{4}} \:\frac{\partial^{\mathrm{2}} \mathrm{z}}{\partial\theta^{\mathrm{2}} }…

Question-197057

Question Number 197057 by ajfour last updated on 07/Sep/23 Commented by ajfour last updated on 07/Sep/23 $${The}\:{ant}\:{has}\:{to}\:{climb}\:{up}\:{the}\:{plane} \\ $$$${and}\:{surmount}\:{the}\:{wall}\:{of}\:{height}\:{c}, \\ $$$${and}\:{descend}\:{then}\:{reach}\:{B}.\:{Find}\:{the} \\ $$$${shortest}\:{length}\:{of}\:{path}. \\ $$…