Moderate -0.8 This is a straightforward application of the chain rule to differentiate a composite function. It requires rewriting the cube root as a power (1+x²)^(1/3), then applying the chain rule mechanically—a routine technique tested early in C3 with no problem-solving element.
$y = \sqrt[3]{1 + x^2} = (1 + x^2)^{1/3}$ | M1, M1, B1 | Do not allow MR for square root their $dy/dx \ne du/dx$ (available for wrong indices) no ft on $\frac{1}{2}$ index
$\frac{dy}{dx} = \frac{1}{3}(1 + x^2)^{-2/3} \cdot 2x$ | |
$= \frac{2}{3}x(1 + x^2)^{-\frac{2}{3}}$ | A1 [4] | cao, mark final answer | oe e.g. $\frac{2x(1 + x^2)^{-\frac{2}{3}}}{3}$, $\frac{2x}{3\sqrt[3]{(1 + x^2)^2}}$, etc but must combine 2 with $1/3$
---