Question

use the four - step definition of the derivative to find $f(x)$ if $f(x)=sqrt{7x}+1$. you may have to simplify your answers to recieve credit.
$f(x + h)=
$f(x + h)-f(x)=
$\frac{f(x + h)-f(x)}{h}=
$f(x)=$

Explanation:

Step1: Find $f(x + h)$

Substitute $x+h$ into $f(x)$:
$f(x + h)=\sqrt{7(x + h)}+1$

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

$$
LATEXBLOCK0
$$

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

$$
LATEXBLOCK1
$$

Step4: Find $f'(x)$

Take the limit as $h
ightarrow0$ of $\frac{f(x + h)-f(x)}{h}$:

$$
LATEXBLOCK2
$$

Answer:

$f(x + h)=\sqrt{7(x + h)}+1$
$f(x + h)-f(x)=\sqrt{7(x + h)}-\sqrt{7x}$
$\frac{f(x + h)-f(x)}{h}=\frac{7}{\sqrt{7(x + h)}+\sqrt{7x}}$
$f'(x)=\frac{\sqrt{7}}{2\sqrt{x}}$