question_answer

If \[y=\frac{{{5}^{x}}}{{{x}^{5}}},\] then find \[\frac{dy}{dx}.\]

Answer:

Given, \[y=\frac{{{5}^{x}}}{{{x}^{5}}}\] Taking logarithm on both sides of Eq. (i), we get \[\log y=\log \,({{5}^{x}})-\log \,({{x}^{5}})\] \[\Rightarrow \]   \[\log y=x\log 5-5\log x\] \[\Rightarrow \]   \[\frac{1}{y}\cdot \frac{dy}{dx}=(log5)\cdot -\frac{5}{x}\] [differentiating both sides w.r.t x] \[\Rightarrow \]   \[\frac{dy}{dx}=y\left( \log 5-\frac{5}{x} \right)\] \[\Rightarrow \]   \[\frac{dy}{dx}=\frac{{{5}^{x}}}{{{x}^{5}}}\left( \log 5-\frac{5}{x} \right)\]