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Category: Differentiation

Prove-that-3-5-2-3-5-2-1-

Question Number 216694 by sniper237 last updated on 16/Feb/25 $${Prove}\:{that}\:\:^{\mathrm{3}} \sqrt{\sqrt{\mathrm{5}}+\mathrm{2}}\:−^{\mathrm{3}} \sqrt{\sqrt{\mathrm{5}}−\mathrm{2}}\:=\mathrm{1} \\ $$ Answered by golsendro last updated on 16/Feb/25 $$\:\mathrm{let}\:\mathrm{x}=\:\sqrt[{\mathrm{3}}]{\sqrt{\mathrm{5}}+\mathrm{2}}−\sqrt[{\mathrm{3}}]{\sqrt{\mathrm{5}}−\mathrm{2}} \\ $$$$\:\sqrt[{\mathrm{3}}]{\sqrt{\mathrm{5}}+\mathrm{2}}−\sqrt[{\mathrm{3}}]{\sqrt{\mathrm{5}}−\mathrm{2}}−\mathrm{x}\:=\:\mathrm{0}\: \\…

without-using-LHopital-rule-evalute-lim-x-0-ln-1-x-sin-x-1-cox-2-x-

Question Number 216638 by Nadirhashim last updated on 13/Feb/25 $$\:\:\boldsymbol{{without}}\:\boldsymbol{{using}}\:\boldsymbol{{LHopital}} \\ $$$$\:\:\:\boldsymbol{{rule}}\:\boldsymbol{{evalute}}\: \\ $$$$\:\:\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\boldsymbol{{ln}}\left(\mathrm{1}−{x}\right)−\boldsymbol{{sin}}\left(\boldsymbol{{x}}\right)\:}{\mathrm{1}−\boldsymbol{{cox}}^{\mathrm{2}} \left(\boldsymbol{{x}}\right)} \\ $$ Commented by MathematicalUser2357 last updated on 13/Feb/25…

u-n-k-n-1-2n-1-k-and-v-n-k-n-2n-1-1-k-show-that-u-n-and-v-n-are-adjacent-use-ln-x-1-x-and-x-ln-1-x-and-show-that-u-n-k-n-1-2n-ln-k-ln-k-1-hence-deduce-

Question Number 215979 by alcohol last updated on 23/Jan/25 $${u}_{{n}} \:=\:\underset{{k}={n}+\mathrm{1}} {\overset{\mathrm{2}{n}} {\sum}}\frac{\mathrm{1}}{{k}}\:{and}\:{v}_{{n}} \:=\:\underset{{k}={n}} {\overset{\mathrm{2}{n}−\mathrm{1}} {\sum}}\frac{\mathrm{1}}{{k}} \\ $$$$\bullet\:{show}\:{that}\:{u}_{{n}} \:{and}\:{v}_{{n}} \:{are}\:{adjacent} \\ $$$${use}\:{ln}\left({x}+\mathrm{1}\right)\:\leqslant\:{x}\:{and}\:{x}\leqslant−{ln}\left(\mathrm{1}−{x}\right)\:{and} \\ $$$$\bullet\:{show}\:{that}\:{u}_{{n}} \:\leqslant\:\underset{{k}={n}+\mathrm{1}}…

Find-the-only-function-that-satisfy-the-expression-below-dy-dx-2-d-2-y-dx-2-

Question Number 215958 by Tawa11 last updated on 22/Jan/25 $$\mathrm{Find}\:\mathrm{the}\:\mathrm{only}\:\mathrm{function}\:\mathrm{that}\:\mathrm{satisfy} \\ $$$$\mathrm{the}\:\mathrm{expression}\:\mathrm{below}: \\ $$$$\:\:\:\:\:\:\:\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)^{\mathrm{2}} \:\:=\:\:\:\frac{\mathrm{d}^{\mathrm{2}} \mathrm{y}}{\mathrm{dx}^{\mathrm{2}} } \\ $$ Answered by Ghisom last updated on…

If-f-x-2-1-x-3-2-u-2-du-find-the-value-of-d-dx-f-1-x-x-2-

Question Number 215809 by efronzo1 last updated on 18/Jan/25 $$\:\:\:\mathrm{If}\:\mathrm{f}\left(\mathrm{x}\right)\:=\:\mathrm{2}\:+\:\underset{\mathrm{1}} {\overset{−\mathrm{x}^{\mathrm{3}} } {\int}}\sqrt{\mathrm{2}+\mathrm{u}^{\mathrm{2}} }\:\mathrm{du}\: \\ $$$$\:\:\:\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\frac{\mathrm{d}}{\mathrm{dx}}\:\left[\mathrm{f}^{−\mathrm{1}} \left(\mathrm{x}\right)\right]_{\mathrm{x}=\mathrm{2}} \\ $$ Answered by mr W last updated…