Easy -1.2 This is a straightforward C1 differentiation question requiring basic algebraic manipulation (rewriting terms with fractional/negative indices) followed by routine application of the power rule. The steps are mechanical with no problem-solving insight needed, making it easier than average.
5. (a) Write \(\frac { 2 \sqrt { } x + 3 } { x }\) in the form \(2 x ^ { p } + 3 x ^ { q }\) where \(p\) and \(q\) are constants.
Given that \(y = 5 x - 7 + \frac { 2 \sqrt { } x + 3 } { x } , \quad x > 0\),
(b) find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), simplifying the coefficient of each term.
Attempt to differentiate either \(2x^p\) with a fractional \(p\), giving \(kx^{p-1}\) (\(k \ne 0\)), (the fraction \(p\) could be in decimal form) or \(3x^q\) with a negative \(q\), giving \(kx^{q-1}\) (\(k \ne 0\)).
Note for (b): It is possible to 'start again' in (b), so the \(p\) and \(q\) may be different from those seen in (a), but note that the M mark is for the attempt to differentiate \(2x^p\) or \(3x^q\).
However, marks for part (a) cannot be earned in part (b).
1st A1ft: If their \(2x^p\), but \(p\) must be a fraction and coefficient must be simplified (the fraction \(p\) could be in decimal form).
2nd A1ft: If their \(3x^q\), but \(q\) must be negative and coefficient must be simplified.
'Simplified' coefficient means \(\frac{a}{b}\) where \(a\) and \(b\) are integers with no common factors. Only a single + or – sign is allowed (e.g. \(- -\) must be replaced by +).
Having +\(C\) loses the B mark.
(a) $p = -\frac{1}{2}$, $q = -1$ | B1, B1 |
(b) $y = 5x - 7 + 2x^{-\frac{1}{2}} + 3x^{-1}$ | —
$\frac{dy}{dx} = 5$ (or $5x^0$) | B1 | ($5x - 7$ correctly differentiated)
Attempt to differentiate either $2x^p$ with a fractional $p$, giving $kx^{p-1}$ ($k \ne 0$), (the fraction $p$ could be in decimal form) or $3x^q$ with a negative $q$, giving $kx^{q-1}$ ($k \ne 0$). | M1 |
$-\frac{1}{2} \times 2x^{\frac{3}{2}} - 1 \times 3x^{-2}$ | = | $-x^{\frac{3}{2}} - 3x^{-2}$ | A1ft, A1ft |
**Total: 4 marks**
**Note for (b):** It is possible to 'start again' in (b), so the $p$ and $q$ may be different from those seen in (a), but note that the M mark is for the attempt to differentiate $2x^p$ or $3x^q$.
However, marks for part (a) cannot be earned in part (b).
**1st A1ft:** If their $2x^p$, but $p$ must be a fraction and coefficient must be simplified (the fraction $p$ could be in decimal form).
**2nd A1ft:** If their $3x^q$, but $q$ must be negative and coefficient must be simplified.
**'Simplified' coefficient** means $\frac{a}{b}$ where $a$ and $b$ are integers with no common factors. Only a single + or – sign is allowed (e.g. $- -$ must be replaced by +).
Having +$C$ loses the B mark.
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5. (a) Write $\frac { 2 \sqrt { } x + 3 } { x }$ in the form $2 x ^ { p } + 3 x ^ { q }$ where $p$ and $q$ are constants.
Given that $y = 5 x - 7 + \frac { 2 \sqrt { } x + 3 } { x } , \quad x > 0$,\\
(b) find $\frac { \mathrm { d } y } { \mathrm {~d} x }$, simplifying the coefficient of each term.\\
\hfill \mbox{\textit{Edexcel C1 2008 Q5 [6]}}