Page 97 - Foundations Of Differential Calculus

80 5. On the Differentiation of Algebraic Functions of One Variable
and so forth. Hence we have

√ dx 2 √ −dx 2 3 √ 1 · 3dx 3
d. x = √ , d . x = √ , d . x = √ , ...,
2 x 4x x 8x 2 x
√ dx 2 √ −2dx 2 3 √ 2 · 5dx 3
3
3
3
d. x = √ , d . x = √ , d . x = √ , ...,
3
3
3
3 x 2 9x x 2 27x 2 x 2
√ dx 2 √ −3dx 2 3 √ 3 · 7dx 3
4
4
4
d. x = √ , d . x = √ , d . x = √ , ....
4
4
4
4 x 3 16x x 3 64x 2 x 3
If we inspect these expressions a bit, we can easily find the differentials,
even without putting the expression into exponential form.
157. If µ is not 1, but some other integer, whether positive or negative, the
differentials can be defined just as easily. Since the second- and higher-order
differentials are defined from the first, using the same law of exponents, we
put down a few of the simpler examples of only first differentials.
√ 3 √ 2 √ 5 √ 3 √ 7 2 √
d.x x = dx x, d.x x = xdx x, d.x x = x dx x, ...,
2 2 2
1 −dx 1 −3dx 1 −5dx
d.√ = √ , d. √ = √ , d. √ = √ , ...,
x 2x x x x 2x 2 x x 2 x 2x 3 x
√ 2dx √ 4 √ √ 5 √
3
3
3
2
2
2
3
3
d. x = √ , d.x x = dx x, d.x x = dx x ,
3
3 x 3 3
√
√ 7 √ 2 3 8 √
3
2 3
3
d.x x = xdx x, d.x x = xdx x, ...,
3 3
1 −dx 1 −2dx 1 −4dx
d. √ = √ , d. √ = √ , d. √ = √ ,
3
3
3
3 x 3x x 3 x 2 3x x 2 x x 3x 2 3 x
1 −5dx 1 −7dx
√ =
√ ,
d. √ = √ , d. 2 3 3 3 ....
3
3
x x 2 3x 2 x 2 x x 3x x
158. From functions of this kind we can find the differentials of all rational
algebraic functions, since each of their terms is a power of x, which we know
how to differentiate. Suppose we have a quantity of the form
p + q + r + s + ··· .
When we substitute x + dx for x we obtain
p + dp + q + dq + r + dr + s + ds + ··· ,