The Chemistry Maths Book, Second Edition

102 Chapter 4Differentiation

3

Many of the rules of differentiation appeared in Leibniz’s 1684 paper on the differential calculus, with the

d–notation and the name calculus differentialisfor the finding of tangents.

Isaac Newton (1642–1727) developed his ideas on the calculus in the year 1665–1666 (Trinity College,

Cambridge, was closed because of the plague); he later maintained that during this time he discovered the

binomial theorem, the calculus, the law of gravitation and the nature of colours. He wrote the first of three

accounts of the calculus in 1669 in De analysi per aequationes numero terminorum infinitas, but published only

in 1711. The first account to be published appeared in 1687 in Philosophiae naturalis principia mathematica,

probably the most influential scientific treatise of all time. In the first edition of the Principia, Newton

acknowledged that Leibniz also had a similar method. By the third edition of 1726, the reference to Leibniz had

been deleted when questions of priority had led to a bitter quarrel between supporters of the two men.

and

(2) (3)

0 Exercise 22

4.6 Differentiation by rule

Whilst all functions can be differentiated from first principles, differentiation is

performed in practice by following a set of rules.

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These rules can be proved from

first principles, as in Examples 4.6 to 4.8. The derivatives of the more important

elementary functions are listed in Table 4.2.

dy

dx

e

x

=

∆

∆

y ∆∆

x

e

xx

x

=++ +(

()

1 )

26

2



∆∆

∆∆

ye x

xx

x

=+++(

() ()

)

23

26



Table 4.2 Differentiation of elementary functions

Type Function Derivative

c 1 = 1 constant c 0

power of xx

a

ax

a− 1

trigonometric sin 1 x cos 1 x

cos 1 x −sin 1 x

tan 1 x sec

2

1 x

exponential e

x

e

x

hyperbolic cosh 1 x sinh 1 x

sinh 1 x cosh 1 x

logarithmic ln 1 x 12 x

The first example in Table 4.2 states that the rate of change of a constant is zero. In

general, if a function is independent of a variable xthen its derivative with respect to

xis zero. Examples of the derivative of a ‘power’ of x, x

a

, have been given in Examples

4.6 fora 1 = 1 − 1 and 4.7 fora 1 = 1122. Other examples, using a variety of notations, are

given in Examples 4.9.