Moderate -0.8 This is a straightforward single application of the chain rule with a simple composition (cube root of a quadratic). It requires only recognizing the outer function (cube root) and inner function (1+x²), then applying the standard chain rule formula—a routine exercise with no problem-solving element beyond pattern recognition.
cao, mark final answer; oe e.g. \(\dfrac{2x(1+x^2)^{-\frac{2}{3}}}{3}\), \(\dfrac{2x}{3\sqrt[3]{(1+x^2)^2}}\) etc but must combine 2 with \(\frac{1}{3}\)
[4]
## Question 6:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $y = \sqrt[3]{1+x^2} = (1+x^2)^{1/3}$ | M1 | $(1+x^2)^{1/3}$; do not allow MR for square root |
| | M1 | Chain rule: their $\frac{dy}{du} \times \frac{du}{dx}$ (available for wrong indices) |
| | B1 | $(1/3)u^{-2/3}$ soi; no ft on $\frac{1}{2}$ index |
| $\dfrac{dy}{dx} = \dfrac{1}{3}(1+x^2)^{-\frac{2}{3}} \cdot 2x$ | | |
| $= \dfrac{2}{3}x(1+x^2)^{-\frac{2}{3}}$ | A1 | cao, mark final answer; oe e.g. $\dfrac{2x(1+x^2)^{-\frac{2}{3}}}{3}$, $\dfrac{2x}{3\sqrt[3]{(1+x^2)^2}}$ etc but must combine 2 with $\frac{1}{3}$ |
| **[4]** | | |
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