# Question 7:

| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{2x^3 - 7}{3\sqrt{x}} = \frac{2x^3}{3\sqrt{x}} - \frac{7}{3\sqrt{x}} = \frac{2}{3}x^{\frac{5}{2}} - \frac{7}{3}x^{-\frac{1}{2}}$ | M1 | Attempts to split the fraction into 2 terms and obtains either $\alpha x^{\frac{5}{2}}$ or $\beta x^{-\frac{1}{2}}$. May be implied by a correct power of $x$ in their differentiation of one of these terms. Beware $\beta x^{-\frac{1}{2}}$ coming from $\frac{2x^3-7}{3\sqrt{x}} = 2x^3 - 7 + 3x^{-\frac{1}{2}}$ |
| $x^n \rightarrow x^{n-1}$ | M1 | Differentiates by reducing power by one for any of their powers of $x$ |
| $\frac{dy}{dx} = 6x + 2x^{-\frac{2}{3}} + \frac{5}{3}x^{\frac{3}{2}} + \frac{7}{6}x^{-\frac{3}{2}}$ | A1 A1 A1 A1 | A1: $6x$. Do not accept $6x^3$. Depends on second M mark only. Award when first seen and isw. A1: $2x^{-\frac{2}{3}}$. Must be simplified. Depends on second M mark only. A1: $\frac{5}{3}x^{\frac{3}{2}}$. Must be simplified but allow e.g. $1\frac{2}{3}x^{1.5}$ or $\frac{5}{3}\sqrt{x^3}$. A1: $\frac{7}{6}x^{-\frac{3}{2}}$. Must be simplified but allow e.g. $1\frac{1}{6}x^{-1\frac{1}{2}}$ or $\frac{7}{6\sqrt{x^3}}$ |

**In an otherwise fully correct solution, penalise the presence of $+ c$ by deducting the final A1**

## Quotient Rule Method:

| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{d\left(\frac{2x^3-7}{3\sqrt{x}}\right)}{dx} = \frac{3\sqrt{x}(6x^2) - (2x^3-7)\frac{3}{2}x^{-\frac{1}{2}}}{(3\sqrt{x})^2}$ | M1 | Uses **correct** quotient rule |
| $= \frac{10x^{\frac{3}{2}} + 7x^{-\frac{1}{2}}}{6x}$ | A1 A1 | A1: Correct first term of numerator and correct denominator. A1: All correct as simplified as shown |

So $\frac{dy}{dx} = 6x + 2x^{-\frac{2}{3}} + \frac{10x^{\frac{3}{2}} + 7x^{-\frac{1}{2}}}{6x}$ scores full marks

**[6 marks total]**

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