Additional Mathematics – Calculus | e-Consult

The derived function is obtained by differentiating term‑by‑term:

\(\displaystyle \frac{dy}{dx} = 6x - 12.\)

At \(x = 4\):

\(\displaystyle \left.\frac{dy}{dx}\right|_{x=4} = 6(4) - 12 = 12.\)

Hence the gradient of the curve at \(x = 4\) is 12. The derived function gives the instantaneous rate of change of \(y\) with respect to \(x\); it tells how rapidly the ordinate changes for a small change in the abscissa, which is exactly the gradient of the tangent to the curve at any point.