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Math Help - Domain & first derivative

  1. #1
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    Domain & first derivative

    <br />
\begin{array}{l}<br />
 f(x) = \sqrt {1 - \sqrt {2 - \sqrt {3 - x} } }  \\ <br />
  \\ <br />
 find \\ <br />
 1)\;domain\;of\;f(x) \\ <br />
 2)\;f'(x) \\ <br />
 \end{array}<br /> <br /> <br />
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  2. #2
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    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".
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  3. #3
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    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
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