Differentiation of Logarithmic Functions

Example 1:

$$ y=ln \,3x,\;\;\; \textup{Find} \;\;\frac{dy}{dx}$$

Solution:

$$ y=ln \,3x$$ $$ \frac{dy}{dx}=\frac{d}{dx} ln \,3x$$ $$ \textup{Let}\; u= 3x$$ $$ \frac{dy}{dx}=\frac{d}{dx} ln\,u$$ $$ \frac{dy}{dx}=\frac{d}{du} ln\,u \frac{du}{dx}$$ $$ \frac{dy}{dx}=\frac{1}u \frac{d}{dx} (3x)$$ $$ \frac{dy}{dx}=\frac{1}{3x}(3)$$ $$ \frac{dy}{dx}=\frac{1}x$$