Question

solve this..... genius... ​

Answer 1

$\underline{\text{Solution :}}$

$\mathrm{Let,\:u=5^{x}\:\&\:v=log_{5}x}$

$\quad\quad\:\;\;\;\;\;\implies \mathrm{v=\frac{log_{e}x}{log_{e}5}}$

$\text{Taking differentials, we get}$

$\mathrm{du=5^{x}\:log_{e}5}$

$\mathrm{dv=\frac{1}{x\:log_{e}5}}$

$\therefore \mathrm{\frac{du}{dv}=\frac{5^{x}\:log_{e}5}{\frac{1}{x\:log_{e}5}}}$

$\to \boxed{\mathrm{\frac{d(5^{x})}{d(log_{5}x)}=x\:5^{x}(log_{e}5)^{2}}}$

$\underline{\text{Formulas :}}$

$\mathrm{1.\:log_{a}X=\frac{log_{e}X}{log_{e}a}}$

$\mathrm{2.\:\frac{d}{dx}(a^{mx})=m\:a^{mx}\:log_{e}a,\:(a>0)\:m}\in \mathbb{R}$

$\mathrm{3.\:\frac{d}{dx}(log_{e}mx)=\frac{1}{x},\:(mx>0)\:m}\in \mathbb{R}$