# How do you differentiate f(t)=(2t)/(2+sqrtt)?

Apr 9, 2018

Using the quotient rule allows for this to be differentiated.

#### Explanation:

Quotient rule:
$f \left(x\right) = g \frac{x}{h \left(x\right)}$
$f ' \left(x\right) = \frac{g ' \left(x\right) h \left(x\right) - h ' \left(x\right) g \left(x\right)}{h {\left(x\right)}^{2}}$

$f \left(t\right) = \frac{2 t}{2 + \sqrt{t}}$

$g \left(t\right) = 2 t$
$h \left(t\right) = 2 + \sqrt{t}$

$g ' \left(t\right) = 2$
$h ' \left(t\right) = \frac{1}{\sqrt{t}}$

This means that
$f ' \left(t\right) = \frac{2 \left(2 + \sqrt{t}\right) - \frac{1}{\sqrt{t}} 2 t}{{\left(2 + \sqrt{t}\right)}^{2}}$
$f ' \left(t\right) = \frac{4 + 2 \sqrt{t} - 2 \sqrt{t}}{{\left(2 + \sqrt{t}\right)}^{2}}$

$f ' \left(t\right) = \frac{4}{{\left(2 + \sqrt{t}\right)}^{2}}$