Moderate -0.8 This is a straightforward C1 differentiation question requiring rewriting terms as powers (x^{1/2} and x^{-1/2}), applying the power rule, and substituting x=8. The algebraic manipulation to express the answer in the form a√2 adds minimal difficulty. This is easier than average as it's purely procedural with no problem-solving or conceptual challenges.
2. Given
$$y = \sqrt { x } + \frac { 4 } { \sqrt { x } } + 4 , \quad x > 0$$
find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(x = 8\), writing your answer in the form \(a \sqrt { 2 }\), where \(a\) is a rational number.
(5)
Decreases any power by 1. Either \(x^{\frac{1}{2}} \rightarrow x^{-\frac{1}{2}}\) or \(x^{-\frac{1}{2}} \rightarrow x^{-\frac{3}{2}}\) or \(4 \rightarrow 0\) or \(x^{\text{their}\,n} \rightarrow x^{\text{their}\,n-1}\) for fractional \(n\)
Correct derivative, simplified or unsimplified including indices. Allow \(\frac{1}{2}-1\) for \(-\frac{1}{2}\) and allow \(-\frac{1}{2}-1\) for \(-\frac{3}{2}\)
\(\sqrt{8} = 2\sqrt{2}\) seen or implied anywhere, including from substituting \(x=8\) into \(y\). May be seen explicitly or implied e.g. \(8^{\frac{3}{2}} = 16\sqrt{2}\) or \(8^{\frac{5}{2}} = 128\sqrt{2}\) or \(4\sqrt{8} = 8\sqrt{2}\)
A1
\(\frac{1}{16}\sqrt{2}\) or \(\frac{\sqrt{2}}{16}\) and allow rational equivalents e.g. \(\frac{32}{512}\). Apply isw.
Total: 5 marks
## Question 2:
$$y = \sqrt{x} + \frac{4}{\sqrt{x}} + 4 = x^{\frac{1}{2}} + 4x^{-\frac{1}{2}} + 4$$
| Working/Answer | Mark | Guidance |
|---|---|---|
| $x^n \rightarrow x^{n-1}$ | M1 | Decreases any power by 1. Either $x^{\frac{1}{2}} \rightarrow x^{-\frac{1}{2}}$ or $x^{-\frac{1}{2}} \rightarrow x^{-\frac{3}{2}}$ or $4 \rightarrow 0$ or $x^{\text{their}\,n} \rightarrow x^{\text{their}\,n-1}$ for fractional $n$ |
| $\frac{dy}{dx} = \frac{1}{2}x^{-\frac{1}{2}} + 4 \times -\frac{1}{2}x^{-\frac{3}{2}}$ $\left(= \frac{1}{2}x^{-\frac{1}{2}} - 2x^{-\frac{3}{2}}\right)$ | A1 | Correct derivative, simplified or unsimplified including indices. Allow $\frac{1}{2}-1$ for $-\frac{1}{2}$ and allow $-\frac{1}{2}-1$ for $-\frac{3}{2}$ |
| $x=8 \Rightarrow \frac{dy}{dx} = \frac{1}{2}8^{-\frac{1}{2}} + 4 \times -\frac{1}{2}8^{-\frac{3}{2}}$ | M1 | Attempts to substitute $x=8$ into their changed expression (even integrated) that is clearly not $y$ |
| $= \frac{1}{2\sqrt{8}} - \frac{2}{(\sqrt{8})^3} = \frac{1}{2\sqrt{8}} - \frac{2}{8\sqrt{8}} = \frac{1}{8\sqrt{2}} = \frac{1}{16}\sqrt{2}$ | B1 | $\sqrt{8} = 2\sqrt{2}$ seen or implied anywhere, including from substituting $x=8$ into $y$. May be seen explicitly or implied e.g. $8^{\frac{3}{2}} = 16\sqrt{2}$ or $8^{\frac{5}{2}} = 128\sqrt{2}$ or $4\sqrt{8} = 8\sqrt{2}$ |
| | A1 | $\frac{1}{16}\sqrt{2}$ or $\frac{\sqrt{2}}{16}$ and allow rational equivalents e.g. $\frac{32}{512}$. Apply isw. |
**Total: 5 marks**
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2. Given
$$y = \sqrt { x } + \frac { 4 } { \sqrt { x } } + 4 , \quad x > 0$$
find the value of $\frac { \mathrm { d } y } { \mathrm {~d} x }$ when $x = 8$, writing your answer in the form $a \sqrt { 2 }$, where $a$ is a rational number.\\
(5)
\hfill \mbox{\textit{Edexcel C1 2017 Q2 [5]}}