Page 100 - Foundations Of Differential Calculus

5. On the Differentiation of Algebraic Functions of One Variable 83
Further examples of this kind are easily treated according to the given laws.

162. If the quantity proposed for differentiation is the power of some func-
tion whose differential we can find, then the preceding rules are sufficient
to find the first differential. Let p be any function of x that is raised to
n
some power and whose differential is dp. Then the first differential of p is
equal to np n−1 dp. From this we obtain the following.
n
I. If y =(a + x) , then
n−1
dy = n (a + x) dx.

2 2 2
II. If y = a − x , then
2 2
dy = −4xdx a − x .
1 2 2 −1
III. If y = = a + x , then
2
a + x 2
−2xdx
dy = 2 .
2
2
(a + x )
√
2
IV. If y = a + bx + cx , then
bdx +2cx dx
dy = √ .
2 a + bx + cx 2

3 4 4 2 4 4 2/3
V. If y = (a − x ) = a − x , then
3
8 3 4 4 − 1 3 −8x dx
dy = − x dx a − x = √ .
4
3
3 3 a − x 4
1 2 − 1 2
VI. If y = √ = 1 − x , then
1 − x 2
3
2 − 2 xdx
dy = xdx 1 − x = √ .
2
(1 − x ) 1 − x 2
3
√
VII. If y = a + bx + x, then
√
√ √
√
dx b (2 x)+ dx dx b +2dx x
dy = = .
√ 2 √ √ 2
3 3
3 a + bx + x 6 x a + bx + x