Derivative of Logarithm over Power
Jump to navigation
Jump to search
Theorem
- $\dfrac \d {\d x} \dfrac {\ln x} {x^n} = \dfrac {1 - n \ln x} {x^{n + 1} }$
Proof
\(\ds \dfrac \d {\d x} \dfrac {\ln x} {x^n}\) | \(=\) | \(\ds \dfrac \d {\d x} x^{-n} \ln x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \ln x \dfrac \d {\d x} x^{-n} + x^{-n} \dfrac \d {\d x} \ln x\) | Product Rule for Derivatives | |||||||||||
\(\ds \) | \(=\) | \(\ds - n x^{-n - 1} \ln x + x^{-n} \dfrac 1 x\) | Primitive of Reciprocal, Primitive of Power | |||||||||||
\(\ds \) | \(=\) | \(\ds -n x^{-n - 1} \ln x + x^{-n - 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {1 - n \ln x} {x^{n + 1} }\) |
$\blacksquare$