8.1 Determine $f'(x)$ from first principles if it is given that $f(x) = -x^2$ - NSC Mathematics - Question 8 - 2022 - Paper 1

Using the power rule:

$f'(x) = \frac{d}{dx}(4x^3) - \frac{d}{dx}(5x^2)$

1. Differentiate:

   $f'(x) = 12x^2 - 10x$

To find the derivative, we use the quotient rule: $D_x \left[ \frac{u}{v} \right] = \frac{u'v - uv'}{v^2}$ Where:

• $u = -6\sqrt{x} + 2$, hence $u' = -\frac{3}{\sqrt{x}}$
• $v = x^4$, hence $v' = 4x^3$

Now applying the quotient rule:

$D_x \left[ -\frac{6\sqrt{x}+2}{x^4} \right] = \frac{(-\frac{3}{\sqrt{x}})x^4 - (-6\sqrt{x}+2)(4x^3)}{(x^4)^2}$

Simplifying this gives:

$= \frac{22x^{3/2} - 8x^3}{x^8} = \frac{22x^{3/2} - 8x^3}{x^8}$