For some constants a and b, find the derivative of (ax^2 + b)^2 - Vidyadip

Question

For some constants a and b, find the derivative of $(ax^2 + b)^2$

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Answer

Here $f (x) = (ax^2 + b)^2 = a^2x^4 + b^2 + 2abx^2$
$\therefore \;f{\text{'}}(x) = \frac{d}{{dx}}[{a^2}{x^4} + {b^2} + 2ab{x^2}]$
$ = {a^2}\frac{d}{{dx}}({x^4}) + \frac{d}{{dx}}({b^2}) + 2ab\frac{d}{{dx}}({x^2})$
$= a^2 \times 4x^3 + 0 + 2ab \times 2x$
$= 4a^2x^3 + 4abx$
$= 4ax(ax^2 + b)$

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