3.2 Basic Characteristics from the definition

$(cf)^{\prime} = \frac{d(cf)}{dx} = c \frac{df}{dx} = cf^{\prime}$

$(f + g)^{\prime} = \frac{d(f + g)}{dx} = \frac{df}{dx} + \frac{dg}{dx} = f^{\prime} + g^{\prime}$

$(f \cdot g)^{\prime} = \frac{d(f \cdot g)}{dx} = \frac{df}{dx} \cdot g + f \cdot \frac{dg}{dx} = f^{\prime} \cdot g + f \cdot g^{\prime}$

$\left(\frac{f}{g}\right)^{\prime} = \frac{d}{dx}\left(\frac{f}{g}\right) = \frac{f^{\prime}g - fg^{\prime}}{g^2}$

$\left(\frac{1}{g}\right)^{\prime} = \frac{d}{dx}\left(\frac{1}{g}\right) = \frac{-g^{\prime}}{g^2}$

$\frac{d}{dx}log_{a}u = \frac{log_{a}e}{u} \frac{du}{dx} \quad a \neq 0, 1$

$\frac{d}{dx} ln(u) = \frac{d}{dx}log_{e}u = \frac{1}{u} \frac{du}{dx}$

$\frac{d}{dx}a^{u} = a^{u} ln(a) \frac{du}{dx}$

$\frac{d}{dx}e^{u} = e^{u} \frac{du}{dx}$

$\frac{d}{dx}u^v = \frac{d}{dx}e^{v \cdot ln(u)} = e^{v \cdot ln(u)} \frac{d}{dx} [v \cdot ln(u)]=vu^{v-1} \frac{du}{dx} + u^v \cdot ln(u) \frac{dv}{dx}$