About this chapter
Pure 4 differentiation handles curves that are not written as y = f(x): parametric curves, implicit curves such as 2x² + xy − y³ = 9, and powers of x like x to the power cos x. It also covers connected rates of change, which the January 2024 examiners found was not well understood by most students.
This chapter covers parametric differentiation, tangents and normals, implicit differentiation, differentiating a to the power x, taking logarithms first, stationary points and vertical tangents, connected rates of change, forming equations from a context, and interpreting a rate. The last question combines differentiation with numerical methods from Pure 3.
The ten questions
- Q1 (6 marks): the normal to 2x² + xy − y³ = 9 at (2, 1), by implicit differentiation.
- Q2 (6 marks): dy/dx for x = 4 sin²t, y = 3 tan t, then an exact tangent at t = π/6.
- Q3 (5 marks): a balloon inflated at a constant rate: how fast its surface area grows.
- Q4 (7 marks): dy/dx for an implicit curve with a constant k, then a stationary point fixes p and k.
- Q5 (6 marks): a curve with 2 to the power x in it, differentiated implicitly, and a tangent's x-intercept.
- Q6 (10 marks): the normal at one point of a parabola meets it again, then a triangle's area.
- Q7 (10 marks): an implicit curve with a gap in its x values, found from dy/dx, and a proof of no stationary points.
- Q8 (10 marks): water flowing into a leaking cone: V in terms of h, a differential equation, and a rate.
- Q9 (10 marks): a cycloid: dy/dx as cot(t/2), a tangent, and where every normal passes.
- Q10 (11 marks): y = x to the power cos x, differentiated with logarithms, then its maximum located numerically.
Key skills tested
Parametric. dy/dx = (dy/dt) ÷ (dx/dt). Use the value of t at the point, not x.
Tangents and normals. The normal's gradient is −1/m. Give the form asked for: ax + by + c = 0 needs integers and the "= 0".
Implicit. y² differentiates to 2y dy/dx, xy to y + x dy/dx by the product rule, and a constant to 0. Collect the dy/dx terms.
Exponentials of other bases. a to the power x differentiates to a to the power x times ln a. For example, 2 to the power x gives 2 to the power x times ln 2.
Taking logarithms. For y = x to the power cos x, take ln of both sides: ln y = cos x ln x. Differentiate, then multiply through by y.
Stationary and vertical. dy/dx = 0 when the numerator is zero; use the curve's equation as well. A vertical tangent comes where the denominator is zero.
Connected rates. Chain the rates together, for example dS/dt = dS/dr × dr/dt. Quote the formula you are using first.
Forming equations. Net rate = rate in − rate out. "Proportional to" needs a constant k, and similar triangles link r and h.
Interpreting. A negative rate means decreasing, and a level stops changing when the rate is zero. Give units.
Worked example
Question 1 from this chapter. The curve C has equation 2x² + xy − y³ = 9, and P(2, 1) lies on C. Find the normal to C at P in the form ax + by + c = 0.
Differentiate implicitly: 4x + (y + x dy/dx) − 3y² dy/dx = 0, using the product rule on xy.
At P(2, 1): 8 + 1 + 2 dy/dx − 3 dy/dx = 0, so dy/dx = 9.
The normal's gradient is −1/9, so y − 1 = −1/9 (x − 2). Multiplying by 9: 9y − 9 = −x + 2, which gives x + 9y − 11 = 0.
Finding the tangent, y = 9x − 17, instead of the normal would earn only the first four marks.
Seen on real papers
Every question in our booklets is original. These are the recent papers where each question type has appeared.
- January 2024, Q3: implicit differentiation with the product rule, then a stationary point that fixes two constants, as in Q4.
- January 2024, Q4: connected rates of change for a cone, found with the chain rule, as in Q3 and Q8.
- January 2024, Q9(a) and (b): a parametric derivative with sec and tan, then a tangent in the form y = mx + c, as in Q2 and Q9.
Where marks are lost
- The product rule on a mixed term. In January 2024 the usual error was differentiating a term in both x and y wrongly, often writing only one of its two parts.
- A constant that survives. Some left the constant k in the derivative, and then struggled to make progress.
- Treating a variable as a constant. In the January 2024 rates question, many treated a length that changes as if it were fixed, and made little further progress.
- Cancelling different angles. A common mistake was cancelling sec terms whose angles were different.
Common questions
Should I write dy/dx = at the start of an implicit line?
No. Differentiate every term first, then rearrange to make dy/dx the subject.
How do I find where a parametric curve has a vertical tangent?
Where dx/dt is zero and dy/dt is not, the gradient is undefined, so the tangent is vertical.
How do I set up a connected rates question?
Write the formula linking the quantities, differentiate it, then chain the rates together, keeping the units.
Where this chapter leads
- Pure 4 Chapter 6: Integration: differential equations formed from rates of change, then solved.
- Pure 4 Chapter 7: Vectors: distances and minimum values, found with vectors instead.