Section $2\mathrm{.}1-$ Derivatives of Logarithmic Functions

Derivative of $lnx$

$\frac{d}{dx}lnx=\frac{d}{dx}ln|x|=\frac{1}{x}$

Derivative of $lng(x)$

$\frac{d}{dx}lng(x)=\frac{1}{g(x)}{g}^{\text{'}}(x)=\frac{g^{\text{'}}(x)}{g(x)}$

Example $1.$ Find $f^{\text{'}}(x)$ for each function using the derivative of $lnx$ and product$/$quotient rules as needed.

$($a$)f(x)=lnx+3x-2$

Ans: $f^{\text{'}}(x)=\frac{1}{x}+3$

$($b$)f(x)=ln3-\frac{1}{2}{x}^2+lnx$

$($c$)f(x)=xlnx+4$

Revrite: $f(x)=(x)(lnx)+4$