Derivatives of Logarithms of Functions

\\ \t $\dfrac{d}{dx}\ln x = \dfrac{1}{x}$ \gap[20] \t #[The derivative of the natural logarithm][La derivada del logaritmo natural]# \\ \t $\dfrac{d}{dx}\log_b x = \dfrac{1}{x \ln b} \quad$ \t #[The derivative of the logarithm with base $b$][La derivada del logaritmo con base $b$]#

#[Applying the chain rule gives][Aplicar la regla de la cadena da]#

\\ \t $\dfrac{d}{dx}\ln \color{#93344}{u} = \dfrac{1}{\color{#93344}{u}}\dfrac{d\color{#93344}{u}}{dx}$ \gap[20] \t #[The derivative of the natural logarithm of a function][La derivada del logaritmo natural de una función]# \\ \t $\dfrac{d}{dx}\log_b \color{#93344}{u} = \dfrac{1}{\color{#93344}{u} \ln b}\dfrac{d\color{#93344}{u}}{dx} \quad$ \t #[The derivative of the logarithm of a function with base $b$][La derivada del logaritmo de una funciócon base $b$]#

#[Examples][Ejemplos]#

\\ $\dfrac{d}{dx}\ln (x^2+x) = \dfrac{1}{x^2+x}\dfrac{d}{dx}[x^2+x]$ \t ${}= \dfrac{2x+1}{x^2+2}$

\\ $\dfrac{d}{dx}\log_3 (x^2+x) = \dfrac{1}{(x^2+x)\ln 3}\dfrac{d}{dx}[x^2+x]$ \t ${}= \dfrac{2x+1}{(x^2+2)\ln 3}$

#[Some for you][Algunos para ti]#

Calculate the given derivative.

$\displaystyle \frac{d}{d%0}\left[\%2\right]$

= BOX*

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$\displaystyle \frac{d}{d%3}\left[\%5\right]$

= BOX*

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