Standard +0.3 This is a straightforward implicit differentiation question requiring the product rule and chain rule (for 3^x). While it involves multiple terms and algebraic manipulation to isolate dy/dx, it follows a standard procedure with no conceptual surprises. The final simplification to the given form adds minor complexity but remains routine for C3/C4 level.
3. A curve \(C\) has equation
$$3 ^ { x } + 6 y = \frac { 3 } { 2 } x y ^ { 2 }$$
Find the exact value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at the point on \(C\) with coordinates (2, 3). Give your answer in the form \(\frac { a + \ln b } { 8 }\), where \(a\) and \(b\) are integers.
Substitutes \(x=2\), \(y=3\) into expression and rearranges for \(\frac{dy}{dx}\)
M1
Do not penalise accuracy errors on method mark
Any correct numerical form e.g. \(\frac{9\ln 3 - \frac{27}{2}}{12}\)
A1
In form \(\frac{p\ln q - r}{s}\)
\(\frac{-9+\ln 729}{8}\) or \(\frac{\ln 729 - 9}{8}\)
A1
Exact answer; accept either equivalent form
# Question 3:
| Answer/Working | Mark | Guidance |
|---|---|---|
| Differentiates $3^x \to 3^x\ln 3$ | B1 | Or $e^{x\ln 3}\to e^{x\ln 3}\ln 3$ |
| Differentiates $6y \to 6\frac{dy}{dx}$ | B1 | |
| Uses product rule on $\frac{3}{2}xy^2$: evidence of $\frac{3}{2}y^2 + kxy\frac{dy}{dx}$ | M1 | $y^2$ must differentiate to $ky\frac{dy}{dx}$, not $2y$; if rule quoted it must be correct |
| Completely correct differential of $\frac{3}{2}xy^2$ | A1 | Need not be simplified |
| $3^x\ln 3 + 6\frac{dy}{dx} = \frac{3}{2}y^2 + 3xy\frac{dy}{dx}$ | — | Full differentiation |
| Substitutes $x=2$, $y=3$ into expression and rearranges for $\frac{dy}{dx}$ | M1 | Do not penalise accuracy errors on method mark |
| Any correct numerical form e.g. $\frac{9\ln 3 - \frac{27}{2}}{12}$ | A1 | In form $\frac{p\ln q - r}{s}$ |
| $\frac{-9+\ln 729}{8}$ or $\frac{\ln 729 - 9}{8}$ | A1 | Exact answer; accept either equivalent form |
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3. A curve $C$ has equation
$$3 ^ { x } + 6 y = \frac { 3 } { 2 } x y ^ { 2 }$$
Find the exact value of $\frac { \mathrm { d } y } { \mathrm {~d} x }$ at the point on $C$ with coordinates (2, 3). Give your answer in the form $\frac { a + \ln b } { 8 }$, where $a$ and $b$ are integers.\\
\hfill \mbox{\textit{Edexcel C34 2016 Q3 [7]}}