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Learning objectives
Core idea
Key formulae
Worked examples
Explore the graph
Common mistakes
Exam tip
Guided practice
Practice questions
Challenge
Summary
Quick quiz
Same topic, different board
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LearnYFM01WFM02 · Further Pure Mathematics 2Maclaurin and Taylor seriesTaylor series, L'Hôpital and Leibniz

Maclaurin and Taylor series · Subtopic 2 of 2

Taylor Series, L'Hôpital and Leibniz

Expanding about $x=a$ instead of $0$ gives the Taylor series $f(x)=\sum\frac{f^{(r)}(a)}{r!}(x-a)^r$ . Series also crack " $\frac00$ " limits (as does L'Hôpital's rule), and Leibniz's theorem gives the $n$ th derivative of a product like a binomial expansion.

Edexcel YFM01 WFM02Edexcel 9FM0 Option≈ 50 minRevision 1 · updated 2026-10-27

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Section 1 of 14

Learning objectives

Find Taylor series about $x=a$ .
Evaluate limits using series or L'Hôpital's rule.
Use Leibniz's theorem for $\frac{d^n}{dx^n}(uv)$ .
Find series solutions of differential equations.

Pause and think

Why can't you write $\ln x$ as a Maclaurin series, and what could you do instead?

Section 2 of 14

Core idea

Start hereZooming in elsewhere

A Maclaurin series zooms in at $x=0$ . But $\ln x$ is undefined there, and sometimes we care about behaviour near $x=2$ or $x=\pi$ . A Taylor series zooms in wherever you like, and the same ideas resolve tricky $\frac00$ limits and nth derivatives.

The big idea

$f(x)=\sum\frac{f^{(r)}(a)}{r!}(x-a)^r$ . For $\frac00$ limits: series, or L'Hôpital $\lim\frac fg=\lim\frac{f'}{g'}$ . Leibniz: $(uv)^{(n)}=\sum\binom nru^{(r)}v^{(n-r)}$ .

1. Taylor series

Worked example $\ln x$ about $x=1$

First three terms.

Goal: Series.
Strategy: Derivatives at $1$ .

$f(1)=0,\ f'(1)=1,\ f''(1)=-1,\ f'''(1)=2$
$f'=\frac1x$ , $f''=-\frac1{x^2}$ , $f'''=\frac2{x^3}$ .
$(x-1)-\frac{(x-1)^2}2+\frac{(x-1)^3}3$
Divide by factorials.

AnswerAs above (the

\ln(1+u)

series with

u=x-1

✓ Check

x=1.1

0.1-0.005+0.000333=0.09533

;

\ln1.1=0.09531

✓.

2. Limits

Why L'Hôpital works

$f(x)\approx f(a)+f'(a)(x-a),\ g(x)\approx g(a)+g'(a)(x-a)$
First-order Taylor approximations.
$\frac{f(x)}{g(x)}\approx\frac{f'(a)(x-a)}{g'(a)(x-a)}$
$f(a)=g(a)=0$ in a $\frac00$ form.
$\to\frac{f'(a)}{g'(a)}$
$(x-a)$ cancels.

Worked example $\lim_{x\to0}\frac{1-\cos x}{x^2}$

Evaluate.

Goal: Limit.
Strategy: L'Hôpital twice.

$\frac00\to\frac{\sin x}{2x}$
Differentiate top and bottom.
$\frac00\to\frac{\cos x}2\to\frac12$
Again.

Answer

\frac12

✓ CheckSeries:

1-\cos x\approx\frac{x^2}2

✓.

3. Leibniz and series solutions

Worked example $\frac{d^3}{dx^3}(x^2e^x)$ at $0$

Evaluate.

Goal: Number.
Strategy: Leibniz with $u=x^2$ .

$\sum_{r=0}^3\binom3r(x^2)^{(r)}e^x$
All derivatives of $e^x$ are $e^x$ .
$=x^2e^x+6xe^x+6e^x+0$
$(x^2)'=2x$ , $(x^2)''=2$ , $(x^2)'''=0$ .
$\text{at }0:\ 6$
Substitute.

Answer

6

✓ Check

x^2e^x=x^2+x^3+\cdots

x^3

coefficient

1=\frac{6}{3!}

✓.

Worked exampleSeries solution of $y'=x+y$ , $y(0)=1$

Up to $x^3$ .

Goal: Series.
Strategy: Differentiate the equation.

$y'(0)=0+1=1$
From the equation.
$y''=1+y'\Rightarrow y''(0)=2;\quad y'''=y''\Rightarrow2$
Differentiate repeatedly.
$y=1+x+\tfrac22x^2+\tfrac26x^3$
Taylor coefficients.

Answer

y=1+x+x^2+\frac13x^3+\cdots

✓ CheckExact solution

y=2e^x-x-1=1+x+x^2+\frac{x^3}3+\cdots

✓.

Common misconceptionQuotient rule L'Hôpital differentiates numerator and denominator separately.

Summary

Taylor about $a$ .
$\frac00$ : series or L'Hôpital.
Leibniz like a binomial.
Series solutions by repeated differentiation.

Indeterminate form: Expressions like $\frac00$ needing further analysis.

HistoryL'Hôpital's rule. The rule appeared in the Marquis de L'Hôpital's 1696 textbook, but he had paid Johann Bernoulli for the result: one of history's earliest maths "purchases".

Fun factTaylor on a trip. Sat-nav predicts your position a moment ahead with a Taylor series in time:

x(t+h)\approx x+hv+\frac12h^2a

Section 3 of 14

Key formulae

Taylor series

f(x)=f(a)+f'(a)(x-a)+\frac{f''(a)}{2!}(x-a)^2+\cdots

L'Hôpital's rule

\frac{f(a)}{g(a)}=\frac00\ \Rightarrow\ \lim_{x\to a}\frac{f(x)}{g(x)}=\lim_{x\to a}\frac{f'(x)}{g'(x)}

Leibniz's theorem

\frac{d^n}{dx^n}(uv)=\sum_{r=0}^n\binom nr u^{(r)}v^{(n-r)}

TrickSeries ODE solutions. Differentiate the equation repeatedly and substitute

x=x_0

each time to get

y''(x_0)

y'''(x_0)

, …, then build the Taylor series.

Section 4 of 14

Worked examples

Example 1 · L'Hôpital0 / 2 steps

Find

\lim_{x\to0}\frac{1-\cos x}{x^2}

Example 2 · Series solution0 / 3 steps

y'=x+y

y(0)=1

. Find the series up to

x^3

Section 5 of 14

Explore the graph

Taylor polynomial of ln x about 1

Loading the interactive graph…

Try this

The cubic matches $\ln x$ closely near $x=1$ .
Far from $1$ (e.g. near $0$ ) the approximation fails.

$\ln x\approx(x-1)-\frac{(x-1)^2}2+\frac{(x-1)^3}3$ .

In the real worldNumerical solvers. Computer methods for ODEs (Euler, Runge–Kutta) are built by matching Taylor series to a chosen order.

Section 6 of 14

Common mistakes

IncorrectL'Hôpital on a non-

\frac00

limit

CorrectCheck the form first

It only applies to

\frac00

\frac\infty\infty

IncorrectQuotient rule in L'Hôpital

CorrectDifferentiate top and bottom separately

It's

\frac{f'}{g'}

, not

\left(\frac fg\right)'

IncorrectLeibniz without binomial coefficients

CorrectInclude

\binom nr

It mirrors

(a+b)^n

lim x→0 of 0/0

L'Hôpital: differentiate both and try again.

Section 7 of 14

Exam tip

Exam tipShow the indeterminate form. Write "this is of the form

\frac00

" before applying L'Hôpital; it's often a required step.

Section 8 of 14

Guided practice

SA-FP-TL-G011 markL1 FoundationNon-calculatorTaylor series

In the Taylor series of

\ln x

about

x=1

, find the coefficient of

(x-1)

Salam Academy Original Practice · Proposed mark scheme

Section 9 of 14

Practice questions

SA-FP-TL-0012 marksL2 StandardNon-calculatorLimits

Find

\displaystyle\lim_{x\to0}\frac{1-\cos x}{x^2}

Salam Academy Original Practice · Proposed mark scheme

SA-FP-TL-0021 markL1 FoundationNon-calculatorLimits

Find

\displaystyle\lim_{x\to0}\frac{\sin3x}{x}

Salam Academy Original Practice · Proposed mark scheme

SA-FP-TL-0032 marksL2 StandardCalculatorTaylor series

Find the coefficient of

(x-2)^2

in the Taylor series of

e^x

about

x=2

(to 4 s.f.).

Salam Academy Original Practice · Proposed mark scheme

SA-FP-TL-0043 marksL3 ExamNon-calculatorLeibniz

Use Leibniz's theorem to find

\frac{d^3}{dx^3}(x^2e^x)

x=0

Salam Academy Original Practice · Proposed mark scheme

More questions on every skill in this subtopic are in the question bank under Practise.

Section 10 of 14

Challenge

SA-FP-TL-0054 marksL4 ChallengingNon-calculatorLeibniz

y'=x+y

y(0)=1

. Find the coefficient of

x^3

in the series solution.

Salam Academy Original Practice · Proposed mark scheme

Section 11 of 14

Summary

Taylor about $a$ : powers of $(x-a)$ .
Limits by series or L'Hôpital.
Leibniz: binomial pattern of derivatives.

Section 12 of 14

Quick quiz

Q1 · SA-FP-TL-Q011 markL1 FoundationNon-calculatorLimits

L'Hôpital's rule can be applied directly to

Salam Academy Original Practice · Proposed mark scheme

Section 13 of 14

Same topic, different board

Specification: IAL Further Mathematics (YFM01) and A level Further Mathematics (9FM0)
Where: Further Pure 1 · Taylor series and Methods in calculus (IAL: WFM02)
Expected: Taylor series about a; series solutions of differential equations; L'Hôpital's rule; Leibniz's theorem for nth derivatives of products.
Calculator: Permitted; show enough working to earn the method marks.
Typical style: Edexcel often opens with a routine part and builds to a "show that" or "prove" part.
Open in: YFM01 · WFM02 9FM0 · Option 9FM0 · Option

Section 14 of 14

Past paper questions

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Salam Academy original content · Content status: reviewed and published. Syllabus mapping: Pearson Edexcel YFM01, 2018 specification (official specification).