Moderate -0.3 This is a straightforward implicit differentiation question requiring students to differentiate term-by-term, apply the product rule to -4xy, collect dy/dx terms, and substitute a point. It's slightly easier than average because it's a single-step application of a standard technique with no conceptual complications, though implicit differentiation itself is a mid-level A-level skill.
EITHER: State \(6y\frac{dy}{dx}\) as the derivative of \(3y^2\)
B1
State \(\pm 4x\frac{dy}{dx} \pm 4y\) as the derivative of \(-4xy\)
B1
Equate attempted derivative of LHS to zero and solve for \(\frac{dy}{dx}\)
M1
M1 conditional on at least one B mark. Allow any combination of signs for second B1.
Obtain answer 2
A1
OR: Obtain correct expression for \(y\) in terms of \(x\)
B1
Differentiate using chain rule
M1
M1 conditional on reasonable attempt at solving the quadratic in \(y\)
Obtain derivative in any correct form
A1
Substitute \(x = 2\) and obtain answer 2 only
A1
Total: 4
## Question 3:
| Answer/Working | Mark | Guidance |
|---|---|---|
| **EITHER:** State $6y\frac{dy}{dx}$ as the derivative of $3y^2$ | B1 | |
| State $\pm 4x\frac{dy}{dx} \pm 4y$ as the derivative of $-4xy$ | B1 | |
| Equate attempted derivative of LHS to zero and solve for $\frac{dy}{dx}$ | M1 | M1 conditional on at least one B mark. Allow any combination of signs for second B1. |
| Obtain answer 2 | A1 | |
| **OR:** Obtain correct expression for $y$ in terms of $x$ | B1 | |
| Differentiate using chain rule | M1 | M1 conditional on reasonable attempt at solving the quadratic in $y$ |
| Obtain derivative in any correct form | A1 | |
| Substitute $x = 2$ and obtain answer 2 only | A1 | **Total: 4** |
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3 Find the gradient of the curve with equation
$$2 x ^ { 2 } - 4 x y + 3 y ^ { 2 } = 3$$
at the point $( 2,1 )$.
\hfill \mbox{\textit{CAIE P3 2004 Q3 [4]}}