2023 H2 Maths Paper 1, Question 1

Differentiating implicitly with respect to $x$, using the chain rule on the right-hand side (with $u = 11 - 5x$, so $\frac{du}{dx} = - 5$):

Multiplying through by $y$ gives a new equation for $\frac{dy}{dx}$ itself:

At $x = 2$: $11 - 5(2) = 1$, so $\ln y = 1^{2} = 1$, giving $y = e$.

Substituting $x = 2$ and $y = e$ into the derivative:

To check your answer: at $x = 2$, $\ln y = 1 \Rightarrow y = e$, and $\frac{dy}{dx} = - 10y(11 - 5x)$ evaluated at $x = 2$ gives $- 10e(1) = - 10e$, consistent ✓

The tangent at $(2,e)$ has gradient $- 10e$:

Adding $e$ to both sides: