Edexcel IAL P1 Differentiation Questions
Gradient of a tangent as a limit; differentiating xⁿ and sums of such terms; gradients, tangents and normals; the second derivative.
- P1 · Pure Mathematics 1
- Non-calculator
- 32 practice questions
What the questions test
The 32 P1 differentiation practice questions cover:
Worked example 1
The tangents to the curve $y = x^2 - 4x$ at the points where $x = 1$ and $x = 3$ intersect at a point $P$. Find the coordinates of $P$.
Mark scheme
- M1 for finding the y-coordinates: at $x=1, y=-3$ and at $x=3, y=-3$.
- M1 for differentiating $\frac{dy}{dx} = 2x - 4$ to find the gradients: $m_1 = -2$ (at $x=1$) and $m_3 = 2$ (at $x=3$).
- M1 for finding the equation of the first tangent: $y - (-3) = -2(x - 1) \implies y = -2x - 1$.
- M1 for finding the equation of the second tangent: $y - (-3) = 2(x - 3) \implies y = 2x - 9$.
- M1 for equating the two tangent equations: $-2x - 1 = 2x - 9 \implies 4x = 8 \implies x = 2$.
- A1 for finding the corresponding y-coordinate $y = -5$, giving $P(2, -5)$.
Worked example 2
The normal to the curve $y = \sqrt{x}$ at the point where $x = 4$ cuts the x-axis at $A$ and the y-axis at $B$. Calculate the exact area of the triangle $AOB$, where $O$ is the origin.
Mark scheme
- M1 for finding the y-coordinate at $x=4$: $y=2$.
- M1 for finding $\frac{dy}{dx} = \frac{1}{2}x^{-\frac{1}{2}}$ and evaluating at $x=4$ to get gradient $m_t = \frac{1}{4}$.
- M1 for finding the normal gradient $m_n = -4$ and its equation: $y - 2 = -4(x - 4) \implies y = -4x + 18$.
- M1 for finding the axis intercepts of the normal: at $y=0, 4x = 18 \implies x = 4.5$ (point $A$) and at $x=0, y=18$ (point $B$).
- M1 for using the area of a triangle formula $\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 4.5 \times 18$.
- A1 for the final area of $40.5$ square units.
FAQ
What differentiation is in Edexcel IAL P1?
Gradient of a tangent as a limit; differentiating xⁿ and sums of such terms; gradients, tangents and normals; the second derivative.
Is P1 a calculator paper?
No. P1 is non-calculator, so every differentiation question has to be done by hand, with exact values where the question asks for them.
How are P1 differentiation questions marked?
With M marks for a correct method, A marks for accurate answers that depend on the M mark before them, and B marks for independent results, as in the worked examples on this page.