# How solve it? Topic: DERIVATE

## Please I need help I do not know how to solve

Jun 1, 2018

$f \left(x\right) = \sqrt{2 x}$

First, let's rewrite the square root as a $1 / 2$ power.

$f \left(x\right) = {\left(2 x\right)}^{\frac{1}{2}}$

Now, we need to recognize that these can be split up as a constant and a variable function.

$f \left(x\right) = {2}^{\frac{1}{2}} \cdot {x}^{\frac{1}{2}}$

When we differentiate, multiplicative constants like the ${2}^{\frac{1}{2}}$ here simply stay on the "outside," that is, we don't do anything to them.

To differentiate ${x}^{\frac{1}{2}}$, we use the power rule, which says that $\frac{d}{\mathrm{dx}} {x}^{n} = n {x}^{n - 1}$.

Then, we see that:

$f ' \left(x\right) = {2}^{\frac{1}{2}} \cdot \left(\frac{1}{2} {x}^{\frac{1}{2} - 1}\right)$

Now simplifying:

$f ' \left(x\right) = {2}^{\frac{1}{2}} / 2 {x}^{- \frac{1}{2}}$

$f ' \left(x\right) = \frac{1}{{2}^{\frac{1}{2}} \cdot {x}^{\frac{1}{2}}}$

$f ' \left(x\right) = \frac{1}{\sqrt{2 x}}$

So at $x = \frac{5}{3}$, the derivative is equal to:

$f ' \left(\frac{5}{3}\right) = \frac{1}{\sqrt{2 \cdot \frac{5}{3}}} = \frac{1}{\sqrt{\frac{10}{3}}} = \sqrt{\frac{3}{10}}$

Jun 1, 2018

$f ' \left(x\right) = \frac{2}{2 \sqrt{2 x}} = \frac{1}{\sqrt{2 x}}$
$f ' \left(\frac{5}{3}\right) = \frac{1}{\sqrt{\frac{10}{3}}} = \frac{\sqrt{3}}{\sqrt{10}}$

#### Explanation:

show below:

$f \left(x\right) = \sqrt{2 x}$

$f ' \left(x\right) = \frac{2}{2 \sqrt{2 x}} = \frac{1}{\sqrt{2 x}}$

The derivative at $x = \frac{5}{3}$ equal

$f ' \left(\frac{5}{3}\right) = \frac{1}{\sqrt{\frac{10}{3}}} = \frac{\sqrt{3}}{\sqrt{10}}$

$\text{Note that}$

$\textcolor{red}{y = \sqrt{x}}$

$\textcolor{red}{y ' = \frac{1}{2 \sqrt{x}} \cdot x '}$

$\textcolor{red}{\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}}$