Question Number 211502 by MathematicalUser2357 last updated on 11/Sep/24
$$\mathrm{If}\:\begin{cases}{{f}\left({x}\right)={x}^{\mathrm{2}} }\\{{g}\left({x}\right)=\mathrm{sin}\:{x}}\end{cases}, \\ $$$$\mathrm{Then}\:\mathrm{find}\:\frac{{df}}{{dg}}. \\ $$
Answered by a.lgnaoui last updated on 11/Sep/24
$$\frac{\mathrm{df}}{\mathrm{dg}}=\frac{\mathrm{df}}{\mathrm{dx}}×\frac{\mathrm{dx}}{\mathrm{dg}}.=\:\:\frac{\mathrm{f}^{'} }{\mathrm{g}'} \\ $$$$\mathrm{f}'=\mathrm{2x}\:\:\:\:\:\:\mathrm{g}'=\mathrm{cos}\:\mathrm{x} \\ $$$$\Rightarrow\:\:\:\:\:\:\:\:\:\frac{\mathrm{df}}{\mathrm{dg}}=\frac{\mathrm{2}\boldsymbol{\mathrm{x}}}{\mathrm{cos}\:\boldsymbol{\mathrm{x}}} \\ $$
Commented by MathematicalUser2357 last updated on 12/Sep/24
$$\mathrm{Can}\:\mathrm{be}\:\mathrm{solved}\:\mathrm{in}\:\mathrm{other}\:\mathrm{way}. \\ $$$$\frac{{df}}{{dg}}=\frac{\mathrm{2}{x}\:{dx}}{\mathrm{cos}\:{x}\:{dx}}=\frac{\mathrm{2}{x}}{\mathrm{cos}\:{x}}=\mathrm{2}{x}\:\mathrm{sec}\:{x}\:\left(\mathrm{No}\:\mathrm{need}\:\mathrm{to}\:\mathrm{transform}\:\frac{\mathrm{1}}{\mathrm{cos}\:{x}}\:\mathrm{to}\:\mathrm{sec}\:{x}\right) \\ $$
Answered by MATHEMATICSAM last updated on 11/Sep/24
$${f}\left({x}\right)\:=\:{x}^{\mathrm{2}} \:\Rightarrow\:\frac{{df}}{{dx}}\:=\:\mathrm{2}{x} \\ $$$${g}\left({x}\right)\:=\:\mathrm{sin}{x}\:\Rightarrow\:\frac{{dg}}{{dx}}\:=\:\mathrm{cos}{x} \\ $$$$\frac{{df}}{{dg}}\:=\:\frac{\frac{{df}}{{dx}}}{\frac{{dg}}{{dx}}}\:=\:\frac{\mathrm{2}{x}}{\mathrm{cos}{x}}\:=\:\mathrm{2}{x}\mathrm{sec}{x} \\ $$
Commented by MathematicalUser2357 last updated on 12/Sep/24
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