8.1 Determine $f'(x)$ from first principles if it is given that $f(x) = 3x^{2}$ - NSC Mathematics - Question 8 - 2021 - Paper 1

To find the derivative, use the power rule and the quotient rule as needed:

1. Differentiate $f(x)$: $f'(x) = \frac{d}{dx}(x^{2}) - \frac{d}{dx}(3) + \frac{d}{dx}\left(\frac{9}{x^{2}}\right)$
2. This yields: $f'(x) = 2x + 0 - 9x^{-3}$ Hence, $f'(x) = 2x - \frac{9}{x^{3}}$

To find the derivative, use the product rule:

Let: $u = \sqrt{x+3}$
$v = \sqrt{x-1}$
Then using the product rule $g'(x) = u'v + uv'$:

1. Calculate $u'$: $u' = \frac{1}{2\sqrt{x+3}}$

2. Calculate $v'$: $v' = \frac{1}{2\sqrt{x-1}}$

3. Now apply the product rule: $g'(x) = \left(\frac{1}{2\sqrt{x+3}}\right)\sqrt{x-1} + \sqrt{x+3}\left(\frac{1}{2\sqrt{x-1}}\right)$ Thus, $g'(x) = \frac{\sqrt{x-1}}{2\sqrt{x+3}} + \frac{\sqrt{x+3}}{2\sqrt{x-1}}$