Question Number 83257 by 09658867628 last updated on 29/Feb/20
$${find}\:{the}\:{derivtive}\:{of}\:{y}=\frac{\mathrm{10}^{{x}} }{{log}_{\mathrm{10}} {x}} \\ $$
Answered by TANMAY PANACEA last updated on 29/Feb/20
$${y}=\frac{{u}}{{v}}\rightarrow\frac{{dy}}{{dx}}=\frac{{v}\frac{{du}}{{dx}}−{u}\frac{{dv}}{{dx}}}{{v}^{\mathrm{2}} } \\ $$$${u}=\mathrm{10}^{{x}} \rightarrow{lnu}={xln}\mathrm{10} \\ $$$$\frac{\mathrm{1}}{{u}}\frac{{du}}{{dx}}={ln}\mathrm{10}\:\boldsymbol{{so}}\:\frac{\boldsymbol{{du}}}{\boldsymbol{{dx}}}=\mathrm{10}^{\boldsymbol{{x}}} \boldsymbol{{ln}}\mathrm{10} \\ $$$$\boldsymbol{{v}}=\boldsymbol{{log}}_{\mathrm{10}} \boldsymbol{{x}}\:\boldsymbol{{so}}\:\:\boldsymbol{{v}}=\frac{\boldsymbol{{lnx}}}{\boldsymbol{{ln}}\mathrm{10}} \\ $$$$\frac{\boldsymbol{{dv}}}{\boldsymbol{{dx}}}=\frac{\mathrm{1}}{\boldsymbol{{xln}}\mathrm{10}} \\ $$$$\frac{\boldsymbol{{dy}}}{\boldsymbol{{dx}}}=\frac{\boldsymbol{{log}}_{\mathrm{10}} \boldsymbol{{x}}.\mathrm{10}^{\boldsymbol{{x}}} \boldsymbol{{ln}}\mathrm{10}−\mathrm{10}^{\boldsymbol{{x}}} .\frac{\mathrm{1}}{\boldsymbol{{xln}}\mathrm{10}}}{\left(\boldsymbol{{log}}_{\mathrm{10}} \boldsymbol{{x}}\right)^{\mathrm{2}} } \\ $$