# Math Help - Domain & first derivative

1. ## Domain & first derivative

$
\begin{array}{l}
f(x) = \sqrt {1 - \sqrt {2 - \sqrt {3 - x} } } \\
\\
find \\
1)\;domain\;of\;f(x) \\
2)\;f'(x) \\
\end{array}

$

2. First, of course, 3- x must be non-negative in order for that square root to be a real number: $3- x\ge 0$ so $x\le 3$. And once we have that we must have $2-\sqrt{3- x}\ge 0$. That is the same as saying that $2\ge \sqrt{3- x}$ or $4\ge 3- x$, $x\ge -1$. So far then, $-1\le x\le 3$. Now, we need to have $1- \sqrt{2- \sqrt{3- x}}\ge 0$. Can you finish that?

As for the derivative, this function is $y= (1- (2- (3- x)^{1/2})^{1/2})^{1/2}$. Use the "power rule" and the "chain rule".

3. $f^{'}(x)={\frac{-1}{8\sqrt{1-\sqrt{2-\sqrt{3-x}}}\sqrt{2-\sqrt{3-x}}\sqrt{3-x}}$,
and then you can simplify the answer