EXAMPLE $6$ Find

f$\text{'}($x$)$

if

f$($x$)=$ ln$(|3$x$|).$

SOLUTION Since

f$($x$)=$

ln$(3$x$)$if x $>0$ln$(3$x$)$if x $<0$

it follows that

f$\text{'}($x$)=$

$1$

$$

ddx

$(3$x$)=$

$$if x $>0$

$1$

$$

ddx

$(3$x$)=$

$$if x $<0$

Thus, f$\text{'}($x$)=$

for all x $0.$

EXAMPLE $6$ Find

f$\text{'}($x$)$

if

f$($x$)=$ ln$(|3$x$|).$

SOLUTION Since

f$($x$)=$

ln$(3$x$)$if x $>0$ln$(3$x$)$if x $<0$

it follows that

f$\text{'}($x$)=$

$1$

$$

ddx

$(3$x$)=$

$$if x $>0$

$1$

$$

ddx

$(3$x$)=$

$$if x $<0$

Thus, f$\text{'}($x$)=$

for all x $0.$