Given $$y = rac{ ext{√}x + rac{4}{ ext{√}x} + 4}{x > 0}$$ find the value of $$\frac{dy}{dx}$$ when $$x = 8$$, writing your answer in the form $$a\sqrt{2}$$, where $$a$$ is a rational number. - Edexcel - A-Level Maths Pure - Question 5 - 2017 - Paper 1

Now we substitute $x = 8$ into the differentiated expression:

$\frac{dy}{dx} = \frac{1}{2\sqrt{8}} - \frac{2}{8^{3/2}}$.

Calculating each term:

1. $\sqrt{8} = 2\sqrt{2}$, so: $\frac{1}{2\sqrt{8}} = \frac{1}{2(2\sqrt{2})} = \frac{1}{4\sqrt{2}}$.

2. For $8^{3/2}$, we have: $8^{3/2} = (\sqrt{8})^3 = (2\sqrt{2})^3 = 8\sqrt{2}$, therefore: $\frac{2}{8^{3/2}} = \frac{2}{8\sqrt{2}} = \frac{1}{4\sqrt{2}}$.

Putting it all together:

$\frac{dy}{dx} = \frac{1}{4\sqrt{2}} - \frac{1}{4\sqrt{2}} = 0$.

Hence, my final answer is:

$\frac{dy}{dx} = 0$.

Since the answer $0$ can also be expressed as $0\sqrt{2}$, it follows that $a = 0$, where $a$ is a rational number.