Standard +0.3 This is a straightforward implicit differentiation question requiring product rule, chain rule, and substitution of a given point. While it involves exponential and trigonometric functions, it follows a standard procedure with no conceptual challenges beyond applying differentiation rules correctly—slightly easier than average for A-level.
3 A curve has equation \(\mathrm { e } ^ { 2 x } \cos 2 y + \sin y = 1\).
Find the exact gradient of the curve at the point \(\left( 0 , \frac { 1 } { 6 } \pi \right)\).
Substitute \(x\)- and \(y\)-values to find value of first derivative
M1
Dependent on at least two terms, with at least one involving \(\frac{dy}{dx}\)
Obtain \(\frac{2}{\sqrt{3}}\) or \(\frac{2}{3}\sqrt{3}\) or exact equivalent
A1
## Question 3:
| Answer | Mark | Guidance |
|--------|------|----------|
| Use product rule to differentiate $e^{2x}\cos 2y$ | M1 | Must be evidence of implicit differentiation |
| Obtain $2e^{2x}\cos 2y - 2e^{2x}\sin 2y\frac{dy}{dx}$ | A1 | |
| Obtain $\left[2e^{2x}\cos 2y - 2e^{2x}\sin 2y\frac{dy}{dx} + \right]\cos y\frac{dy}{dx} = 0$ | B1 | |
| Substitute $x$- and $y$-values to find value of first derivative | M1 | Dependent on at least two terms, with at least one involving $\frac{dy}{dx}$ |
| Obtain $\frac{2}{\sqrt{3}}$ or $\frac{2}{3}\sqrt{3}$ or exact equivalent | A1 | |
---
3 A curve has equation $\mathrm { e } ^ { 2 x } \cos 2 y + \sin y = 1$.\\
Find the exact gradient of the curve at the point $\left( 0 , \frac { 1 } { 6 } \pi \right)$.\\
\hfill \mbox{\textit{CAIE P2 2022 Q3 [5]}}