For some constant $a$ and $b$ , find the derivative of the following functions:
$(a x^{2} + b)^{2}$ .

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Solution

Let $f (x) = {(a x^{2} + b)}^{2}$
$\Rightarrow f (x) = a^{2} x^{4} + 2 a b x^{2} + b^{2}$
$∴ f^{^{'}} (x) = \frac{d}{d x} (a^{2} x^{4} + 2 a b x^{2}) = a^{2} \frac{d}{d x} (x^{4}) + 2 a b \frac{d}{d x} (x^{2}) + \frac{d}{d x} (b^{2})$
$= a^{2} (4 x^{3}) + 2 a b (2 x) + b^{2} (0)$
$= 4 a^{2 x^{3}} + 4 a b x = 4 a x (a x^{2} + b)$

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