Question Number 203448 by hardmath last updated on 19/Jan/24
$$\mathrm{y}\:=\:\mathrm{sin3x}−\mathrm{ln}\left(\mathrm{3x}+\mathrm{1}\right)+\mathrm{3}^{\mathrm{2}\boldsymbol{\mathrm{x}}} \\ $$$$\mathrm{find}:\:\:\:\mathrm{y}^{'} \:=\:? \\ $$
Commented by hardmath last updated on 19/Jan/24
$$ \\ $$find the derivative of the function
Answered by mr W last updated on 19/Jan/24
$${y}'=\mathrm{3}\:\mathrm{cos}\:\mathrm{3}{x}−\frac{\mathrm{3}}{\mathrm{3}{x}+\mathrm{1}}+\mathrm{3}^{\mathrm{2}{x}} ×\mathrm{2}\:\mathrm{ln}\:\mathrm{3} \\ $$
Commented by hardmath last updated on 19/Jan/24
$$\mathrm{thank}\:\mathrm{you}\:\mathrm{dear}\:\mathrm{professor} \\ $$
Answered by MrGHK last updated on 19/Jan/24
$$\boldsymbol{\mathrm{y}}^{'} =\frac{\boldsymbol{\mathrm{d}}}{\boldsymbol{\mathrm{dx}}}\left(\mathrm{3}\boldsymbol{\mathrm{x}}\right).\frac{\boldsymbol{\mathrm{d}}}{\boldsymbol{\mathrm{du}}}\left(\boldsymbol{\mathrm{sin}}\left(\boldsymbol{\mathrm{u}}\right)\right)−\frac{\boldsymbol{\mathrm{d}}}{\boldsymbol{\mathrm{dx}}}\left(\mathrm{3}\boldsymbol{\mathrm{x}}+\mathrm{1}\right).\frac{\boldsymbol{\mathrm{d}}}{\boldsymbol{\mathrm{du}}}\left(\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{u}}\right)\right)+\frac{\boldsymbol{\mathrm{d}}}{\boldsymbol{\mathrm{dx}}}\left(\mathrm{2}\boldsymbol{\mathrm{xln}}\left(\mathrm{3}\right)\right).\frac{\boldsymbol{\mathrm{d}}}{\boldsymbol{\mathrm{du}}}\boldsymbol{\mathrm{e}}^{\boldsymbol{\mathrm{u}}} \\ $$$$\boldsymbol{\mathrm{y}}^{'} =\mathrm{3}\boldsymbol{\mathrm{cos}}\left(\mathrm{3}\boldsymbol{\mathrm{x}}\right)−\frac{\mathrm{3}}{\mathrm{3}\boldsymbol{\mathrm{x}}+\mathrm{1}}+\mathrm{2}\boldsymbol{\mathrm{ln}}\left(\mathrm{3}\right)\mathrm{3}^{\mathrm{2}\boldsymbol{\mathrm{x}}} \\ $$