Page 101 - Foundations Of Differential Calculus

84 5. On the Differentiation of Algebraic Functions of One Variable
1
VIII. If y = √ , then since
2
x + a − x 2
−xdx
2 2
d. a − x = √ ,
2
a − x 2
we have

√
√
2
−dx +(xdx) a − x 2 xdx − dx a − x 2
2
dy = √ 2 √ 2 √ ,
2 2 = 2 2 2 2
x + a − x x + a − x a − x
or
√ 3
2 2
dx x − a − x
dy = 2 √ .
2
2
2
(2x − a ) a − x 2

3

4 1 3 2 2
IX. If y = 1 − √ + (1 − x ) ,welet
x
1 2
3
2
√ = p and (1 − x ) = q;
x

4 3
since y = (1 − p + q) ,wehave
−3dp +3dq
dy = √ .
4
4 1 − p + q
From previous work we have
−dx −4xdx
dp = √ and dq = √ .
3
2x x 3 1 − x 2
When these results are substituted, we have
√
√

3 2
(3dx) (2x x) − (4xdx) 1 − x
dy = .

4 1 3 2
2
4 1 − √ + (1 − x )
x
In a similar way, by substituting individual letters for terms to be com-
posed, we can easily find the differentials of this kind of function.
163. If the quantity that is to be differentiated is the product of two or
more functions of x whose differentials are known, the most convenient
method for finding the differential is as follows. Let p and q be functions of
x with differentials dp and dq already known. When we substitute x+dx for