Find the derivative of the following
(i) $x^{x^{n}} + x^{n^{x}}$
(ii) $l o g \sqrt{x^{2} + 1}$

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Solution

$(i) y = x^{x^{n}} + x^{n^{x}}$

$y^{'} = x^{n} (x^{x^{n} - 1}) n x^{n - 1} + n^{x} x^{n^{x} - 1} n^{x} ln n$

$(i i) y = l o g \sqrt{x^{2} + 1}$

$y^{'} = \frac{1}{\sqrt{x^{2} + 1}} \times \frac{2 x}{2 \sqrt{x^{2} + 1}}$
$y^{'} = \frac{x}{x^{2} + 1}$

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