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Category: Differentiation

Given-a-function-y-f-x-where-f-1-x-5-x-5-8-x-5-Find-slope-of-the-curve-y-f-x-at-x-1-

Question Number 124007 by liberty last updated on 30/Nov/20 $$\:{Given}\:{a}\:{function}\:{y}={f}\left({x}\right)\:{where}\:{f}^{−\mathrm{1}} \left(\frac{{x}+\mathrm{5}}{{x}−\mathrm{5}}\right)=\frac{\mathrm{8}}{{x}+\mathrm{5}} \\ $$$${Find}\:{slope}\:{of}\:{the}\:{curve}\:{y}={f}\left({x}\right)\:{at}\:{x}=\mathrm{1}\:. \\ $$ Answered by john_santu last updated on 30/Nov/20 $${f}^{−\mathrm{1}} \left(\frac{{x}+\mathrm{5}}{{x}−\mathrm{5}}\right)=\frac{\mathrm{8}}{{x}+\mathrm{5}}\:\Leftrightarrow\:{f}\left(\frac{\mathrm{8}}{{x}+\mathrm{5}}\right)=\frac{{x}+\mathrm{5}}{{x}−\mathrm{5}} \\…

nice-calculus-prove-that-0-x-x-1-x-4-x-4-dx-2-1-4-pi-is-Golden-ratio-

Question Number 123961 by mnjuly1970 last updated on 29/Nov/20 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:…{nice}\:\:\:{calculus}… \\ $$$$\:\:{prove}\:\:{that}:: \\ $$$$ \\ $$$$\:\:\:\:\Omega=\:\int_{\mathrm{0}} ^{\:\infty} \left(\frac{{x}^{\varphi} }{\left({x}+\mathrm{1}\right)\sqrt{{x}^{\mathrm{4}\varphi} +{x}^{\mathrm{4}} }}\right){dx} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\overset{???} {=}\:\frac{\varphi\Gamma^{\mathrm{2}} \left(\frac{\mathrm{1}}{\mathrm{4}}\right)}{\:\sqrt{\pi}}…

nice-calculus-find-a-series-representation-for-the-following-integral-0-1-cos-h-xln-x-dx-

Question Number 123898 by mnjuly1970 last updated on 29/Nov/20 $$\:\:\:\:\:\:\:\:\:…\:{nice}\:\:{calculus}… \\ $$$$\:\:{find}\:\:{a}\:{series}\:{representation} \\ $$$$\:{for}\:\:{the}\:{following}\:{integral}\::: \\ $$$$\:\:\:\:\phi=\int_{\mathrm{0}} ^{\:\mathrm{1}} {cos}\left({h}\left({xln}\left({x}\right)\right){dx}\right. \\ $$$$ \\ $$ Answered by mindispower…

Among-all-triangles-in-the-first-quadrant-formed-by-the-x-axis-the-y-axis-and-tangent-lines-to-the-graph-of-y-3x-x-2-what-is-the-smallest-possible-area-

Question Number 123817 by benjo_mathlover last updated on 28/Nov/20 $${Among}\:{all}\:{triangles}\:{in}\:{the}\:{first} \\ $$$${quadrant}\:{formed}\:{by}\:{the}\:{x}−{axis}, \\ $$$${the}\:{y}−{axis}\:{and}\:{tangent}\:{lines}\:{to} \\ $$$${the}\:{graph}\:{of}\:{y}\:=\:\mathrm{3}{x}−{x}^{\mathrm{2}} ,\:{what}\:{is} \\ $$$${the}\:{smallest}\:{possible}\:{area}? \\ $$ Answered by MJS_new last…

Within-the-interval-0-x-2pi-find-the-critical-points-of-f-x-sin-2-x-sin-x-1-Identify-the-open-interval-on-which-f-is-increasing-and-decreasing-Find-the-function-s-local-and-absolute-extreme-v

Question Number 123812 by benjo_mathlover last updated on 28/Nov/20 $${Within}\:{the}\:{interval}\:\mathrm{0}\leqslant{x}\leqslant\mathrm{2}\pi\:,{find}\: \\ $$$${the}\:{critical}\:{points}\:{of}\: \\ $$$${f}\left({x}\right)=\:\mathrm{sin}\:^{\mathrm{2}} {x}−\mathrm{sin}\:{x}−\mathrm{1}\:.\:{Identify} \\ $$$${the}\:{open}\:{interval}\:{on}\:{which}\:{f}\:{is}\: \\ $$$${increasing}\:{and}\:{decreasing}\:.\:{Find} \\ $$$${the}\:{function}'{s}\:{local}\:{and}\:{absolute} \\ $$$${extreme}\:{values}. \\ $$…

Let-f-x-be-differentiable-function-and-g-x-be-the-inverse-of-f-x-when-lim-x-2-f-x-x-3-x-2-4-1-2-then-dg-x-dx-x-8-

Question Number 123752 by liberty last updated on 27/Nov/20 $${Let}\:{f}\left({x}\right)\:{be}\:{differentiable}\:{function} \\ $$$${and}\:{g}\left({x}\right)\:{be}\:{the}\:{inverse}\:{of}\:{f}\left({x}\right). \\ $$$${when}\:\underset{{x}\rightarrow\mathrm{2}} {\mathrm{lim}}\:\left(\frac{{f}\left({x}\right)−{x}^{\mathrm{3}} }{{x}^{\mathrm{2}} −\mathrm{4}}\right)=\frac{\mathrm{1}}{\mathrm{2}}\:{then}\:\frac{{dg}\left({x}\right)}{{dx}}\mid_{{x}=\mathrm{8}} \:=? \\ $$ Commented by benjo_mathlover last updated…

y-x-1-3-x-1-2-d-2-y-

Question Number 123667 by sahnaz last updated on 27/Nov/20 $$\mathrm{y}=\left(\mathrm{x}+\mathrm{1}\right)^{\mathrm{3}} \left(\mathrm{x}−\mathrm{1}\right)^{\mathrm{2}} \:\:\:\:\:\:\:\:\mathrm{d}^{\mathrm{2}} \mathrm{y}=? \\ $$ Answered by mathmax by abdo last updated on 27/Nov/20 $$\mathrm{y}\left(\mathrm{x}\right)=\left(\mathrm{x}+\mathrm{1}\right)^{\mathrm{3}}…