Implicit Differentiation — Mark Scheme Patterns

Mark Allocation Overview

Question Type	Total Marks	M marks	A marks	B marks
Find $\frac{dy}{dx}$	3	1	2	0
Find $\frac{d^2y}{dx^2}$	5	2	3	0
Values at a point (combined)	8	3	5	0

Pattern: First Derivative $\frac{dy}{dx}$ (3 marks)

• M1: Differentiate implicitly — must show evidence of chain rule on $y$ terms $\left(\frac{d}{dx}f(y) = f'(y)\frac{dy}{dx}\right)$ and product rule on $xy$ terms
• A1: Correct differentiated equation (all terms correct)
• A1: Correct expression for $\frac{dy}{dx}$ (fully simplified)

扣分点

• 对 $y^2$ 求导只得到 $2y$ 而非 $2y\frac{dy}{dx}$ → 扣 M1
• 乘积法则 $xy$ 只得到 $y$ 而非 $y + x\frac{dy}{dx}$ → 扣 M1

Pattern: Second Derivative $\frac{d^2y}{dx^2}$ (5 marks)

• M1: Attempt quotient rule on $\frac{dy}{dx}$ expression
• M1: Correctly differentiate $y$ terms in numerator/denominator (with $\frac{dy}{dx}$)
• A1: Correct unsimplified expression for $\frac{d^2y}{dx^2}$
• A1: Substitute expression for $\frac{dy}{dx}$ correctly
• A1: Correct simplified $\frac{d^2y}{dx^2}$ in terms of $x$ and $y$ only

简化技巧

终态表达式应不含 $\frac{dy}{dx}$，仅含 $x$ 和 $y$。可利用原方程进一步化简。

Pattern: Values at Specific Points (8 marks total)

Part (a) — Find $\frac{dy}{dx}$ (3 marks):

• M1: Implicit differentiation
• A1: Correct derivative
• A1: Expression for $\frac{dy}{dx}$

Part (b) — Tangent/Normal (2 marks):

• M1: Substitute coordinates into $\frac{dy}{dx}$
• A1: Correct equation of tangent or normal

Part (c) — Second derivative at point (3 marks):

• M1: Differentiate $\frac{dy}{dx}$ again
• M1: Substitute coordinates and $\frac{dy}{dx}$ value
• A1: Correct numerical answer

关键技巧

求 $\frac{d^2y}{dx^2}$ 在特定点的值时，可在代入数值后再求导，而非先化简一般表达式再代入。有时代入后计算更简单（因为 $\frac{dy}{dx}=0$ 时大量项消失）。

Common MS Coding

Code	Meaning
M1	Method — implicit differentiation shown
A1	Accuracy — correct derivative expression
A1	Accuracy — correct simplified form
ft	Follow-through — accept error from part (a)
AG	Answer given — show full working