Derivatives of Logarithms of Functions
\\ \t $\dfrac{d}{dx}(e^x) = e^x$ \gap[20] \t
\\ \t $\dfrac{d}{dx}(b^x) = b^x \ln b \quad$ \t
#[Applying the chain rule gives][Aplicar la regla de la cadena da]#
\\ \t $\dfrac{d}{dx}[e^{\color{blue}{u}}] = e^{\color{blue}{u}}\dfrac{d\color{blue}{u}}{dx}$ \gap[20] \t
\\ \t $\dfrac{d}{dx}[b^{\color{blue}{u}}] = b^{\color{blue}{u}} \ln b\dfrac{d\color{blue}{u}}{dx} \quad$ \t
#[Examples][Ejemplos]#
\\ $\dfrac{d}{dx}\left[e^{x^2+x}\right] = e^{x^2+x}\dfrac{d}{dx}[x^2+x]$ \t ${}= (2x+1)e^{x^2+x}$
\\ $\dfrac{d}{dx}\left[b^{x^2+x}\right] = b^{x^2+x}\ln b\dfrac{d}{dx}[x^2+x]$ \t ${}= (2x+1)b^{x^2+x}\ln b$