Leibniz Notation: Alternative Derivative Expression by Gottfried Leibniz, Schemes and Mind Maps of History

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Leibniz Notation

dy 2

ax bx c

dx

Newton Leibniz

The last years of his life - from 1709 to 1716 - were embittered by the long controversy

with John Keill, Newton, and others, as to whether he had discovered the differential

calculus independently of Newton's previous investigations, or whether he had derived

the fundamental idea from Newton, and merely invented another notation for it. The

controversy occupies a place in the scientific history of the early years of the eighteenth

century quite disproportionate to its true importance, but it materially affected the

history of mathematics in western Europe.

Gottfried Wilhelm Leibnitz (1646 - 1716)

On the accession in 1714 of his master, George I., to the throne of England, Leibnitz was thrown aside as a useless tool; he was forbidden to come to England; and the last two years of his life were spent in neglect and dishonour. He was overfond of money and personal distinctions; was unscrupulous, as perhaps might be expected of a professional diplomatist of that time; but all who once came under the charm of his personal presence remained sincerely attached to him.

He also held eminent positions in diplomacy, philosophy and literature.

From `A Short Account of the History of Mathematics' (4th edition, 1908) by W. W. Rouse Ball.

Leibniz Notation

Liebniz Notation is an alternative way of expressing derivatives to f'(x) , g'(x) , etc.

If y is expressed in terms of x then the derivative is written as dy/ dx.

eg y = 3x^2 - 7x so^

dy/ dx =^ 6x^ -^ 7.

Regardless of the notation, the meaning of the result is the same:

• the rate of change of the function
• the gradient of the tangent at a given point

“dee y by dee x”

If function is given as y=, or x=, t= etc. we use Leibniz notation.

Example 1

Solution:

1. Prepare for differentiation

2 3 Q 9 R 15 R − = −

(2 1) ( 3 1) 2 9 ( 3) 15

dQ R R dR

− − − =  − − 

2. “Multiply by power then

reduce power by 1”

3. Tidy up 18 45 4 dQ R R dR

− = +

4

dQ R dR R

If Q = 9R^2 - 15

R^3

find dQ/dR!

Extra bit for “FIZZY SYSTS” or even Physicists.

Newton’s 2ndLaw of Motion

s = ut + 1 / 2 at^2 where s = distance & t = time.

Finding ds/dt gives us a “diff in dist”  “diff in time”

ie speed or velocity

so ds/dt = u + at

but ds/dt = v so we get v = u + at

and this is Newton’s 1st Law of Motion

• Heinemann , p.100, EX 6I, Q 2-