+

+ Yny

+

+

+

Yn we have

Differentiation of a Sum of Functions. 14. If u =

where Y + y2 + Y3

Yu, Y2 Y3, . ... Yn, denote any functions whatever of x, then

du_dy, dy, dy dy,
dx dx dx

dx

dx In fact, taking any simultaneous values t', u', y'v 4'2, y',, ... y'n of x, U, Yu Y2, 43, ...

Y. + Y2 + Y3 +

+ Yn
u' = y + '+ y's +
'

+ y'me
u' – u _ 4,- y, _Y':-Y2+ : – Y3 + ... +
– uy'

Y'

whence

X'
c'

x
du

ду, ду,

dx Сх ex and therefore, proceeding to the limit,

du dy, dy, dyz
da dx dx dx

U =

1

3

+

2

+ -X

+4=1

1

ម

Hence the differential coefficient of the sum of any number of functions is equal to the sum of the differential coefficients of the functions taken separately.

Differentiation of the Product of two Functions. 15. If u = y.yy, where y, and y, are any functions of x, then :

, du

dx We have c', u', yn y'a, denoting simultaneous values of 2, u, y, Y2

YY., u = yı'ya', u-ury'15,- yya = y, (y', - y.) + y (y', - y)+(4,-y)(- y), du

dy, dy, .dy,

, Yi

Ex Sr

dy2 dx

dy, + Y2 dx

whence, proceeding to the limit, that is, equating dx to zero, and therefore, y, and y, being supposed to be continuous functions of x, putting also dy, and dy, each equal to zero, we have

du

Yi
dx
du = y, dy, + y, dy

, Hence the differential coefficient of the product of two functions is equal to the sum of the products of each function multiplied by the differential coefficient of the other.

or

Differentiation of the Ratio of two Functions. 16. If u = %, then

:
,
du 1

y
Y2

dx dx Taking a', u', yi'ya', to denote simultaneous values of X, U, Yu, Y , we have

dy,

1

dy2 - Yi

dx y

2

1

1

dy

Ex Y Y

Y, 8x

{y, (y'- y) - y, (yı - y)},
yy.
du

dy, whence

y2 Y2Y2

8x and therefore, proceeding to the limit, that is, equating a' – x or dx to zero, we get, observing that y; becomes y,

du 1

dy

dy, dx y

1 du

(ydy, - y,dy.).

y Y2 Hence, to differentiate the Ratio of two functions, we have the following rule: Multiply the denominator by the differential coefficient of the numerator, and the numerator by the differential coefficient of the denominator: subtract the latter product

Ya dx

Y, ax

2

from the former : this difference divided by the square of the denominator is the differential coefficient of the Ratio.

,

Differentiation of the Product of any number of Functions. 17. If u = y;-YYz... Yn, the product of n functions of x, then 1 du 1 dy.. 1 dy,

1 dyz dx y dx y, dx y, dr

dix

ขน

dx

n

2.

3

,

n

+

+

+

u
n

adding these equations together, cancelling terms which are common to both sides of the resulting equation, and observing

that u,

=

Yu, we have

18. If

y

be a function of x, in which case x will also be a function of y, then

dy dx

1.

Let x', y', be simultaneous values of x,y; then it is evident that

x'
1,

1;
x' x y' - y
and that, consequently, proceeding to the limit,

y' - y

dy dx

dx dy

dy dx

1.

dx dy

a

u

y x

Differentiation of a Function of a Function. 19. If u be a function of y, and y a function of x, then

du du dy

dx dy dx For we have, by common algebra, taking a', y', u', as simultaneous values of x, y, U,

u u 6 g - 9
X

y
and therefore, proceeding to the limit,

du du dy

dx dy dx COR. If u be a function of y., y, of yz, y, of yz,. .. . and y..

of x, it is manifest that we may prove in the same way that

du du dy, dy, dyn-ı. dy.
dx dy, dy, dy,

dx Take for instance three y's; then

u
u

y'' - , Y- : Y: - Y3
x'
y' -% ya - y. y - y Y:

x' and, in the limit,

du
du dy,

dy, dy,
do dy, dy, dy, dx

a

dyn

x

2

Differentiation of a Function of two Functions. 20. Let u = f(y1, y), where f(y,, y) denotes any function whatever of y, and y, each of the quantities y, and y, being a

Y9

ya - Ya

u

2

=

dy,

i

function of a third quantity c. Let y., 42, u, become yi', ya', u', when x becomes x': then

U = f(y,, Y.), u = f(yi, y,), u – u = f(yı, y,) - f(y,, Y.)

= f(y;', y.) - f(y., y) + f(yi, y:) - f(y's y.), u flyi', y)-f(y1, ) y-y, f(y'sy'))-f(y',y) ya-Y2

x' y - y

Now in the limit, when a differs from « less than by any assignable magnitude, u Du yi - y, dy, ya

Y.' - y_dy,
x
dx X'
dx X'

dx
fly', y) - f(y,, y) _df (4,, y) du
y' - y

dy,'
and, first replacing yı' by Yı, and then yı' by yz,
f(y', y:') – f(yi, y)-f(yı, y,') - f(y1, y) af(y1,

y) du ya - y

ya - y

dy

dy, Du du dy, du dy

,, , hence

(1). dx In this equation it is very important to observe that the numerators of the two fractions

du du

dy dy, although represented by the same symbol du, are essentially different, the numerator of the former corresponding to the ultimate value of the increment

flyi, y.)-f(y, y), and the numerator of the latter to the ultimate value of the increment.

f(yı, y)-f(yı, y). The numerator of the fraction

Du
dä

;

1

+

dx dy,

dy, dr

dy, dx