About this chapter
Differentiation in Pure 1 is less about the rule, which most students learn quickly, and more about what happens before and after it. Before: rewriting roots and fractions as powers of x. After: using the gradient to build a tangent or normal, find an unknown constant, or locate another point on the curve with the same gradient.
This chapter covers differentiating any power of x, second derivatives, the gradient as the limit of a chord, rates of change in context, and tangents and normals. Stationary points belong to Pure 2, so they are not used here. The longer questions include a tangent that meets a cubic again, the condition f′(x) = f′′(x), a tangent of given gradient that leads to a quadratic in x², and a triangle formed by a tangent, a normal and the y-axis.
The ten questions
- Q1 (5 marks): dy/dx and the second derivative of 4x³ − 6/x² + 5√x.
- Q2 (5 marks): rewrite (3x² − 4)/(2√x) as a sum of powers of x, then differentiate it.
- Q3 (5 marks): the gradient of a chord of y = x² − 3x is 5 + h, and what that says about the tangent.
- Q4 (6 marks): a tank model with fractional powers of t: dV/dt, its exact value at t = 3, and what it means.
- Q5 (6 marks): verify a point on a cubic, then the tangent and the normal there.
- Q6 (8 marks): the tangent where a cubic crosses the y-axis, and the other point where it meets the curve.
- Q7 (9 marks): two points on a cubic with parallel tangents, then show that the midpoint of RP is also on the curve.
- Q8 (9 marks): f′(2) = f′′(2) fixes k, then the other point where f′(x) = f′′(x), and a normal.
- Q9 (8 marks): the line y = 5x + c touches y = x³ + 2/x, giving an equation in x² and then c exactly.
- Q10 (10 marks): the tangent and normal to y = (x−4)√x at (4, 0), and the area of the triangle they make with the y-axis.
Key skills tested
Differentiating powers of x. Multiply by the power, then reduce the power by one; this works for negative and fractional powers too. For example, 4/x² differentiates to −8/x³.
Rewrite first. Expand brackets and split fractions into separate powers of x before differentiating. For example, (x² + 1)/√x becomes x√x + 1/√x.
Gradients and rates of change. Substitute into dy/dx, showing the powers exactly. A negative dV/dt means V is decreasing.
Second derivative. Differentiate dy/dx again to get the second derivative, written d²y/dx² or f′′(x).
Gradient as a limit. Find the gradient of the chord from x to x + h, then let h tend to 0. For example, a chord gradient of 5 + h tends to 5.
Tangents. The gradient at x = a is f′(a), and the tangent passes through (a, f(a)).
Normals. The normal is perpendicular to the tangent, so its gradient is −1/m.
A given gradient. Solve f′(x) = m. The result may be a hidden quadratic, for example in x².
Equations with unknowns. A given gradient, or a condition such as f′ = f′′ at a point, gives an equation for the unknown constant.
Worked example
Question 5 from this chapter. The curve C has equation y = x³ − 4x² + 7. (a) Verify that P(2, −1) lies on C. (b) Find the tangent at P in the form y = mx + c. (c) Find the normal at P in the form ax + by + c = 0.
(a) At x = 2: 8 − 16 + 7 = −1, so P lies on C.
(b) dy/dx = 3x² − 8x, which is 12 − 16 = −4 at x = 2. The tangent is y + 1 = −4(x−2), so y = −4x + 7.
(c) The normal gradient is the negative reciprocal of −4, which is 1/4. Then y + 1 = ¼(x−2); multiplying by 4 gives 4y + 4 = x − 2, so x − 4y − 6 = 0.
Seen on real papers
Every question in our booklets is original. These are the recent papers where each question type has appeared.
- January 2023, Q1: differentiating negative and fractional powers, then an exact gradient in surd form, as in Q1 and Q4(b).
- June 2023, Q4: writing an expression with roots as powers of x before differentiating, as in Q2.
- January 2023, Q10(c): the tangent where a cubic crosses the y-axis, as in Q6.
- October 2024, Q9(c): another point on a cubic with the same gradient, as in Q7.
- June 2024, Q7: f′(x) = f′′(x) at a given x fixes k, then the second such point, as in Q8.
- January 2025, Q5: a tangent of known gradient leading to a quadratic in x², as in Q9.
- June 2024, Q10(a): a normal where fractional powers must be evaluated by hand, like Q10.
Where marks are lost
- Adding + c. In January 2023 a significant number differentiated correctly and then added a constant, which only belongs to integration.
- Fractional powers. In June 2024 a significant proportion could not evaluate the fractional powers when finding a gradient. Work them out as roots: 4 to the power 3/2 is (√4)³ = 8.
- Not substituting the given x. With f′(x) = f′′(x) at x = 5, many in June 2024 never substituted 5, or reached for the discriminant instead.
- A line instead of a point. Asked for another point with the same gradient in October 2024, the most common error was finding the tangent line instead of solving f′(x) equal to that gradient.
- The final square root. In January 2025, after solving a quadratic in x², some forgot to take the square root, and others used a negative x when x had to be positive.
Common questions
Are stationary points in Pure 1?
No. Maximum and minimum points are in Pure 2, so this chapter stays with gradients, tangents, normals and second derivatives.
What is the difference between a tangent and a normal?
Both pass through the same point on the curve. The tangent has gradient f′(a); the normal is perpendicular to it, with gradient −1/f′(a).
Do I need differentiation from first principles?
You need the idea that the gradient of a chord tends to the gradient of the tangent as h tends to 0, as tested in Q3.
Where this chapter leads
- Pure 1 Chapter 9: Integration: reversing differentiation to find f(x) from f′(x).
- Pure 2 Chapter 7: Differentiation: stationary points, increasing and decreasing functions, and optimisation.