Page 114 - Foundations Of Differential Calculus

5. On the Differentiation of Algebraic Functions of One Variable 97
are algebraic functions of x, provided that y is such a function. Furthermore,
if z is also an algebraic function of x, all finite expressions made up of
differentials of any order of y, z, and dx, such as
4
2
3
d y , d y , dx d y ,
2
3 2
2
d z dz d y dy d z
are all likewise algebraic functions of x.
177. Since the first differential of any algebraic function of x can now
be found by the method given, using the same method we can investigate
the second- and higher-order differentials. If y is any algebraic function of
x, from differentiation we have dy = pdx, and we note the value of p.If
2 2
we differentiate again and obtain dp = qdx, then d y = qdx , supposing
that dx is constant. In this way we have defined the second differential.
When we differentiate q, so that dq = rdx, we have the third differential
3 3
d y = rdx . In this way we investigate the differentials of higher order,
and since the quantities p, q, r,... are all algebraic functions of x, the
given laws for differentiation are sufficient. Therefore, we have continuous
differentiation. If we omit the dx in the differentiation of y, we obtain the
value dy/dx = p, which is again differentiated and divided by dx to obtain
2
2
q = d y/dx . Each time we divide by dx, since everywhere the differential
3
3
dx is omitted. In a similar way we obtain r = d y/dx , and so forth.
2
a
I. Let y = ; find the first- and higher-order differentials.
2
a + x 2
First we differentiate and divide by dx to obtain
2
dy −2a x
= 2
2
2
dx (a + x )
and then
2 4 2 2
d y = −2a +6a x ,
2
2
dx 2 (a + x ) 3
3
2 3
4
d y = 24a x − 24a x ,
2
2
dx 3 (a + x ) 4
2 4
4
6
4 2
d y = 24a − 240a x + 120a x ,
2
2
dx 4 (a + x ) 5
5 6 4 3 2 5
d y = −720a x + 2400a x − 720a x ,
2
2
dx 5 (a + x ) 6
and so forth.