Question

use the four - step definition of the derivative to find $f(x)$ if $f(x)=3x - 6$.
$f(x + h)=
$f(x + h)-f(x)=
$\frac{f(x + h)-f(x)}{h}=
find $f(x)$ by determining $lim_{h
ightarrow0}\frac{f(x + h)-f(x)}{h}=$
question help: video

Explanation:

Step1: Find $f(x + h)$

Substitute $x+h$ into $f(x)$:
$f(x + h)=3(x + h)-6=3x+3h - 6$

Step2: Calculate $f(x + h)-f(x)$

\[

$$\begin{align*} f(x + h)-f(x)&=(3x + 3h-6)-(3x - 6)\\ &=3x+3h - 6-3x + 6\\ &=3h \end{align*}$$

\]

Step3: Compute $\frac{f(x + h)-f(x)}{h}$

$\frac{f(x + h)-f(x)}{h}=\frac{3h}{h}=3$

Step4: Determine the limit as $h

ightarrow0$
$f'(x)=\lim_{h
ightarrow0}\frac{f(x + h)-f(x)}{h}=\lim_{h
ightarrow0}3 = 3$

Answer:

$f(x + h)=3x+3h - 6$
$f(x + h)-f(x)=3h$
$\frac{f(x + h)-f(x)}{h}=3$
$f'(x)=3$