WAEC 2016 · Paper 2 · Q4

(a)
Find, from first principles, the derivative of $\dfrac{5}{x^2 + 3}$ with respect to $x$ .

Worked solution (try it first)

$f(x + h) - f(x) = \dfrac{5}{(x + h)^2 + 3} - \dfrac{5}{x^2 + 3}$ .
Over one denominator, the top is $5(x^2 + 3) - 5[(x + h)^2 + 3] = 5(x^2 - x^2 - 2xh - h^2)$
$= -10xh - 5h^2$ .
So $f(x + h) - f(x) = \dfrac{-10xh - 5h^2}{[(x + h)^2 + 3](x^2 + 3)}$ .
Divide by $h$ : $\dfrac{-10x - 5h}{[(x + h)^2 + 3](x^2 + 3)}$ .
Let $h \to 0$ : $f'(x) = -\dfrac{10x}{(x^2 + 3)^2}$ .