AI TutorBetaTopicalsYearlysNotesDashboardPricing
Notes/Mathematics/Paper 2/Differentiation
CAIEAS Level9709§2.4

Differentiation

Five new standard derivatives, the product and quotient rules, and two new ways of describing a curve — parametric and implicit.

32 min read 3 sub-topics
115
question parts
2021–2025 · 37 papers
12 marks
per paper
≈ 25% of the paper
2.5/3
avg difficulty
demanding
#2
most examined
of 6 topics by marks

Differentiation answers one question: how steep is this curve, here? Every application in Paper 1 is that question in disguise.

The mechanics are small — one rule for powers, one chain rule, and that is the whole of the syllabus content. What carries the 462 marks is knowing which application a question is asking for. The single most recorded technique across five years of Paper 1 is "set the first derivative equal to zero" — that is the stationary-point move, and it appears in a fifth of all differentiation parts.

Note what is not here: the syllabus says only an informal understanding of a limit is expected, differentiation from first principles is not required, and points of inflexion are not included. If a method feels more advanced than the four applications below, it is not being asked for.

Before you start you should be able to

  • Laws of indices, including 1x2=x−2\dfrac{1}{x^2} = x^{-2} and x=x1/2\sqrt{x} = x^{1/2}

  • Gradients, and m1m2=−1m_1m_2 = -1 (see Coordinate Geometry)

  • Solving quadratic equations and inequalities

By the end of this page you can

  • Explain the gradient of a curve as the limit of chord gradients, and use f′(x)\mathrm{f}'(x), dydx\dfrac{\mathrm{d}y}{\mathrm{d}x}, f′′(x)\mathrm{f}''(x), d2ydx2\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2}

  • Differentiate xnx^n for any rational nn, together with sums, differences and constant multiples

  • Use the chain rule on composite functions such as (2x−1)−2(2x-1)^{-2}

  • Find the equations of the tangent and the normal at a point

  • Find where a function is increasing or decreasing

  • Locate stationary points and determine their nature with the second derivative

  • Solve connected rates-of-change problems with dAdt=dAdr×drdt\dfrac{\mathrm{d}A}{\mathrm{d}t} = \dfrac{\mathrm{d}A}{\mathrm{d}r} \times \dfrac{\mathrm{d}r}{\mathrm{d}t}

  • (Paper 2) Differentiate ex\mathrm{e}^x, ln⁡x\ln x, sin⁡x\sin x, cos⁡x\cos x and tan⁡x\tan x, and their composites

  • (Paper 2) Use the product and quotient rules

  • (Paper 2) Differentiate a parametrically defined curve with dydx=dy/dtdx/dt\dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{\mathrm{d}y/\mathrm{d}t}{\mathrm{d}x/\mathrm{d}t}

  • (Paper 2) Differentiate implicitly, and find a tangent or normal from the result

01

The standard derivatives

Syllabus requirement · §2.4

“

use the derivatives of eˣ, ln x, sin x, cos x, tan x, together with constant multiples, sums, differences and composites.

”

Paper 1 gave you one derivative — the power rule. Paper 2 adds five more, and they simply have to be known. Everything after this section is about combining them.

Learn these five
ddxex=ex\frac{\mathrm{d}}{\mathrm{d}x}\mathrm{e}^x = \mathrm{e}^x

unchanged — the reason e is the natural base

ddxln⁡x=1x\frac{\mathrm{d}}{\mathrm{d}x}\ln x = \frac{1}{x}

the missing n = −1 case from Paper 1

ddxsin⁡x=cos⁡x\frac{\mathrm{d}}{\mathrm{d}x}\sin x = \cos x

radians only

ddxcos⁡x=−sin⁡x\frac{\mathrm{d}}{\mathrm{d}x}\cos x = -\sin x

note the minus sign

ddxtan⁡x=sec⁡2x\frac{\mathrm{d}}{\mathrm{d}x}\tan x = \sec^2 x

and sec x = 1/cos x (see Trigonometry §06)

Radians, always

ddxsin⁡x=cos⁡x\dfrac{\mathrm{d}}{\mathrm{d}x}\sin x = \cos x is only true when xx is in radians. In degrees the derivative picks up a factor of π180\dfrac{\pi}{180}, which is why all calculus with trigonometric functions is done in radians.

With the chain rule

Each of these composes exactly as (ax+b)n(ax+b)^n did in §02: differentiate the outside, then multiply by the derivative of the inside.

ddxeax+b=a eax+b\frac{\mathrm{d}}{\mathrm{d}x}\mathrm{e}^{ax+b} = a\,\mathrm{e}^{ax+b}

the a comes out front

ddxln⁡(ax+b)=aax+b\frac{\mathrm{d}}{\mathrm{d}x}\ln(ax+b) = \frac{a}{ax+b}

derivative of the inside, over the inside

ddxsin⁡(ax+b)=acos⁡(ax+b)\frac{\mathrm{d}}{\mathrm{d}x}\sin(ax+b) = a\cos(ax+b)

and similarly for cos and tan

Worked example

0 marks

Differentiate: (a) y=e3x−1y = \mathrm{e}^{3x-1} (b) y=ln⁡(2x2+5)y = \ln(2x^2+5) (c) y=sin⁡3xy = \sin^3 x

Show full working
  1. 1

    (a) Outside gives e3x−1\mathrm{e}^{3x-1}, inside gives 33: dydx=3e3x−1\frac{\mathrm{d}y}{\mathrm{d}x} = 3\mathrm{e}^{3x-1}

  2. 2

    (b) Derivative of the inside over the inside: dydx=4x2x2+5\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{4x}{2x^2+5}

  3. 3

    (c) Read sin⁡3x\sin^3 x as (sin⁡x)3(\sin x)^3 — a power on the outside, sin⁡x\sin x inside: dydx=3(sin⁡x)2×cos⁡x=3sin⁡2xcos⁡x\frac{\mathrm{d}y}{\mathrm{d}x} = 3(\sin x)^2 \times \cos x = 3\sin^2 x\cos x

    sin³x is a composite in disguise. Rewriting it with the bracket makes the chain rule obvious.

Answer

(a) 3e3x−13\mathrm{e}^{3x-1} (b) 4x2x2+5\dfrac{4x}{2x^2+5} (c) 3sin⁡2xcos⁡x3\sin^2 x\cos x

Practise the standard derivativesReal past-paper questions · Derivatives of e^x, ln x, sin x, cos x, tan x and composites

The rest of this note

Checking your access…

Can you do all of these?

  • Write down all five standard derivatives without looking

  • Differentiate e3x−1\mathrm{e}^{3x-1}, ln⁡(2x2+5)\ln(2x^2+5) and sin⁡3x\sin^3 x

  • Differentiate x2ln⁡xx^2\ln x and find its stationary point exactly

  • State the quotient rule with the numerator in the right order

  • Find dydx\dfrac{\mathrm{d}y}{\mathrm{d}x} for a parametric curve, and locate a vertical tangent

  • Differentiate x2+y2=xy+7x^2 + y^2 = xy + 7 implicitly and find the gradient at (3,1)(3, 1)

Now do the questions
115 real Paper 2 parts from 2021–2025, sorted by difficulty, with mark schemes
Previous
Trigonometry
Next
Integration