If $f(x) = \sin^{-1} \left ( \frac{4^{x + \frac{1}{2}}}{1 + 2^{4x}} \right )$, which of the following is not the derivative of f(x)?

Put $4^x = \tan \theta$. Then $\theta = \tan^{-1}(4^x)$

$\therefore f(x) = \sin^{-1} \left ( \frac{2 \tan \theta}{1 + \tan \theta} \right )$

$= \sin^{-1} (\sin 2 \theta)$

$= 2 \theta$

$= 2 \tan^{-1} (4^x)$

$\therefore f'(x) = 2 \times \frac{1}{1 + (4^x)^2} \times 4^x \log 4$

$= \frac{2.4^x \cdot \log 4}{1 + 4^{2x}}$ ....(a)

$= \frac{2.4^x \cdot 2 \log 2}{1 + 4^{2x}}$

$= \frac{4^{x + 1} \cdot \log 2}{1 + 4^{2x}}$ ...(b)

$= \frac{(2^{2})^{x + 1} \cdot \log 2}{1 + 2^{4x}}$

$= \frac{2^{2^(x + 1)} \cdot \log 2}{1 + 2^{4x}}$ ....(d)