Derivatives of Logarithms
1 $\dfrac{d}{dx}\ln x = \dfrac{d}{dx}\ln \big|x\big| = \dfrac{1}{x}$ \t
\\ 2. $\dfrac{d}{dx}\log_b x = \dfrac{d}{dx}\log_b \big|x\big| =\dfrac{1}{x \ln b} \quad$ \t
#[Chain rule versions][Vesiones regla de cadena]#
1 $\dfrac{d}{dx}\ln \color{indianred}{u} = \dfrac{d}{dx}\ln \color{indianred}{\big|u\big|} = \dfrac{1}{\color{indianred}{u}}\dfrac{d\color{indianred}{u}}{dx}$ \gap[20] \t
\\ 2. $\dfrac{d}{dx}\log_b \color{indianred}{u} = \dfrac{d}{dx}\log_b \color{indianred}{\big|u\big|} = \dfrac{1}{\color{indianred}{u} \ln b}\dfrac{d\color{indianred}{u}}{dx} \quad$ \t
#[Examples][Ejemplos]#
1. $\dfrac{d}{dx}(\log_5\big|x\big|) = \dfrac{1}{x\ln\,5}$ \gap[5] \t
\\ 2. $\dfrac{d}{dx}(5\,\ln\big|x\big|) = 5\cdot\dfrac{1}{x} = \dfrac{5}{x}$ \gap[5] \t
\\ 3. $\dfrac{d}{dx}(-4\,\log_{20} x) = -4\cdot\dfrac{1}{x\ln\,20}$ \gap[5] \t
\\ $\qquad {}= -\dfrac{4}{x\ln\,20}$
\\ 4. $\dfrac{d}{dx}\left(x^2\,\ln\big|x\big|\right) = 2x\,\ln x+ x^2\dfrac{1}{x}$ \gap[5] \t
\\ $\qquad {}= 2x\,\ln x+x$