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Category: Differentiation

lets-say-we-have-a-function-f-x-The-tangent-line-at-x-n-makes-an-angle-with-the-x-axis-What-is-the-rate-of-change-of-the-angle-as-we-change-the-value-of-x-Suppose-f-x-x-2-

Question Number 3707 by Filup last updated on 19/Dec/15 $$\mathrm{lets}\:\mathrm{say}\:\mathrm{we}\:\mathrm{have}\:\mathrm{a}\:\mathrm{function}\:{f}\left({x}\right). \\ $$$$\mathrm{The}\:\mathrm{tangent}\:\mathrm{line}\:\mathrm{at}\:{x}={n}\:\mathrm{makes}\:\mathrm{an} \\ $$$$\mathrm{angle}\:\theta\:\mathrm{with}\:\mathrm{the}\:{x}−\mathrm{axis}. \\ $$$$ \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{rate}\:\mathrm{of}\:\mathrm{change}\:\mathrm{of}\:\mathrm{the}\:\mathrm{angle} \\ $$$$\mathrm{as}\:\mathrm{we}\:\mathrm{change}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{x}? \\ $$$$ \\ $$$$\mathrm{Suppose}\:{f}\left({x}\right)={x}^{\mathrm{2}} \\…

To-conserve-rice-from-a-cooperative-there-is-a-foil-of-area-A-24cm-2-The-cooperatives-want-to-build-a-barn-in-the-shape-of-a-parallelopiped-square-rectangle-without-cover-What-should-be-the-symmetr

Question Number 69211 by Mikael last updated on 21/Sep/19 $${To}\:{conserve}\:{rice}\:{from}\:{a}\:{cooperative}\:{there}\:{is} \\ $$$${a}\:{foil}\:{of}\:{area}\:{A}=\mathrm{24}{cm}^{\mathrm{2}} .\:{The}\:{cooperatives} \\ $$$${want}\:{to}\:{build}\:{a}\:{barn}\:{in}\:{the}\:{shape}\:{of}\:{a}\:{parallelopiped} \\ $$$${square}\:{rectangle}\:{without}\:{cover}. \\ $$$${What}\:{should}\:{be}\:{the}\:{symmetrical}\:{dimensions}\:{for} \\ $$$${the}\:{volume}\:{to}\:{be}\:{maximum}? \\ $$ Terms of…

nice-calculus-prove-that-n-1-2n-n-1-2n-1-1-2-m-n-

Question Number 134662 by mnjuly1970 last updated on 06/Mar/21 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:….{nice}\:\:{calculus}… \\ $$$$\:\:\:\:{prove}\:\:{that}:: \\ $$$$\:\:\:\boldsymbol{\phi}=\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\zeta\left(\mathrm{2}{n}\right)}{\left({n}+\mathrm{1}\right)\left(\mathrm{2}{n}+\mathrm{1}\right)}=\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\:\:\:\:\:\:\:\:\:…{m}.{n}… \\ $$ Answered by Dwaipayan Shikari last…

Question-69046

Question Number 69046 by Rio Michael last updated on 18/Sep/19 Commented by mathmax by abdo last updated on 18/Sep/19 $$\left.\mathrm{4}\right){we}\:{have}\:\:{y}={xarctanx}\:\Rightarrow{y}^{'} ={arctanx}\:+\frac{{x}}{\mathrm{1}+{x}^{\mathrm{2}} }\:\Rightarrow \\ $$$$\left({x}+{x}^{\mathrm{3}} \right){y}^{'}…

y-ax-n-y-y-a-x-x-n-dy-dx-alim-x-0-x-x-n-x-n-x-Show-that-dy-dx-anx-n-1-

Question Number 3370 by Filup last updated on 12/Dec/15 $${y}={ax}^{{n}} \\ $$$${y}+\delta{y}={a}\left({x}+\delta{x}\right)^{{n}} \\ $$$$\vdots \\ $$$$\frac{{dy}}{{dx}}={a}\underset{\delta{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\left({x}+\delta{x}\right)^{{n}} −{x}^{{n}} }{\delta{x}} \\ $$$$ \\ $$$$\mathrm{Show}\:\mathrm{that}\:\frac{{dy}}{{dx}}={anx}^{{n}−\mathrm{1}} \\ $$…

nice-calculus-prove-that-n-1-1-1-n-4-cosh-2pi-cos-2pi-4pi-2-

Question Number 134442 by mnjuly1970 last updated on 03/Mar/21 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{nice}\:…..\:{calculus}… \\ $$$$\:\:\:\:{prove}\:{that}\::: \\ $$$$\:\:\:\:\:\:\:\:\:\boldsymbol{\phi}=\underset{{n}=\mathrm{1}} {\overset{\infty} {\prod}}\left(\mathrm{1}+\frac{\mathrm{1}}{{n}^{\mathrm{4}} }\right)=\frac{{cosh}\left(\mathrm{2}\pi\right)−{cos}\left(\mathrm{2}\pi\right)}{\mathrm{4}\pi^{\mathrm{2}} } \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:………………….. \\ $$ Answered by Dwaipayan…