Zderivujte:
   $$
      y=\sqrt{x + \sqrt{x + \sqrt x}}
   $$
   \reseni
   \begin{equation*}\begin{split}
      y'(x)&={\left(x + \sqrt{x + \sqrt x}\right)}' \cdot \frac12
      \cdot {\left(x + \sqrt{x + \sqrt x}\right)}^{-\frac12} =\\
      &=\frac12\Biggl(1 +{\biggl(\sqrt{x + \sqrt x}\biggr)}'\Biggr)
      {\left(x + \sqrt{x + \sqrt x}\right)}^{-\frac12} =\\
      &=\frac12\Biggl(1 + {\bigl(x+\sqrt x\bigr)}' \cdot \frac12
      \cdot {\bigl(x + \sqrt x\bigr)}^{-\frac12}\Biggr)
      {\left(x + \sqrt{x + \sqrt x}\right)}^{-\frac12} =\\
      &=\frac1{2\sqrt{x + \sqrt{x + \sqrt x}}}
      \left(1 + \frac{1 + \frac1{2\sqrt x}}
      {2\sqrt{x+\sqrt x}}\right)=\\
      &=\frac{1 + \frac{1 + \frac1{2\sqrt x}}
      {2\sqrt{x+\sqrt x}}}{2\sqrt{x + \sqrt{x + \sqrt x}}}=
      \frac{1 + \frac{2\sqrt x + 1}{4\sqrt x \sqrt{x + \sqrt x}}}
      {2\sqrt{x + \sqrt{x + \sqrt x}}}=
      \frac{\frac{1 + 2\sqrt x + 4\sqrt x \sqrt{x + \sqrt x}}
      {4\sqrt x \sqrt{x + \sqrt x}}}
      {2\sqrt{x + \sqrt{x + \sqrt x}}}=\\
      &=\frac{1 + 2\sqrt x + 4\sqrt x \sqrt{x + \sqrt x}}
      {8\sqrt x \sqrt{x + \sqrt x}\sqrt{x + \sqrt{x + \sqrt x}}}
   \end{split}\end{equation*}
