use the four - step definition of the derivative to find $f(x)$ if $f(x)=3x - 6$.\n$f(x + h)=\n$f(x +…

use the four - step definition of the derivative to find $f(x)$ if $f(x)=3x - 6$.\n$f(x + h)=\n$f(x + h)-f(x)=\n$\frac{f(x + h)-f(x)}{h}=\nfind $f(x)$ by determining $lim_{h\rightarrow0}\frac{f(x + h)-f(x)}{h}=$\nquestion help: video

use the four - step definition of the derivative to find $f(x)$ if $f(x)=3x - 6$.\n$f(x + h)=\n$f(x + h)-f(x)=\n$\frac{f(x + h)-f(x)}{h}=\nfind $f(x)$ by determining $lim_{h\rightarrow0}\frac{f(x + h)-f(x)}{h}=$\nquestion help: video

Answer

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 $h\rightarrow0$

$f'(x)=\lim_{h\rightarrow0}\frac{f(x + h)-f(x)}{h}=\lim_{h\rightarrow0}3 = 3$

Answer:

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