# Question #9d0c5

Feb 10, 2018

Choice e

#### Explanation:

The definition of the derivative is

$f ' \left(x\right) = {\lim}_{h \to 0} \frac{f \left(x + h\right) - f \left(x\right)}{h}$

We are given $f \left(x\right) = 3 {x}^{2} - 2 \sqrt{x}$

We also need to find $f \left(x + h\right)$ which can be done by substituting $x + h$ for $x$ in the function

So...

$f \left(x + h\right) = 3 {\left(x + h\right)}^{2} - 2 \sqrt{x + h}$

Given this we can rewrite the definition of the derivative and find the expression that will find $f ' \left(x\right)$

Thus,

$f ' \left(x\right) = {\lim}_{h \to 0} \frac{\left(3 {\left(x + h\right)}^{2} - 2 \sqrt{x + h}\right) - \left(3 {x}^{2} - 2 \sqrt{x}\right)}{h}$

So given the choices, this will be choice e