196 
DERIVATIVES AND INTEGRALS 
[VI 
4 - then 
= + [ 1 
2/ £&£ y\ dx y<i dx~ "' y n ofo? ' 
In particular if y — 2 W , - - ( ^f . 
y dx z dx 
5. Employ Theorem (9) and Ex. 3 to show that the differential coefficient 
of x m is mx m ~ 1 for all rational values of rn. 
[First suppose m = l/2’, where q is an integer. If y=x x i q , x=y q and 
dyldx = \j{dxjdy) = (l/q)y 1 ~ q ={l/q)x( l l q )~ 1 . Next, if m=p/q, y = z r> , where 
z=x l/q , and the proof may be completed by means of the result of Ex. 3.] 
6. If y—arc sin x, x=siny and 
dy = i j = 1 + 1 
dx l\dyj cos y ~ v /(l — x T )' 
The positive sign must be chosen if cos y is positive, and in particular if 
— \iv<iy<i Jtt. 
97. Standard forms. We shall now investigate more 
systematically the forms of the derivatives of a few of the 
simplest types of functions, 
A. Polynomials. If </>(#) = a 0 x n + a^x 11 - 1 + ... + a n , then 
4>' (x) = na 0 a' n_1 + (n — 1) a 1 # n ~ 2 + ... + a n _j. 
It is sometimes more convenient to use for the standard form of a 
polynomial of degree n in x what is known as the binomial form, 
or the form with binomial coefficients, viz. 
a 0 x n + a^x 11 - 1 + Q a 2 x n ~ 2 +...+«„ 
which differs from the first form in that the coefficients are taken 
to be Oj, ^^a 2 , ••• instead of a ly a 2 , In this case 
(f)’(x) = n Ja 0 « n - 1 + ^ a 1 x n ~ 2 + ^ 1 2 ^ a 2 x n ~ 3 + ...+a»_ 1 |. 
The binomial form of (f)(x) is often written symbolically as 
(«0, , •••, a n ^x, l) n j 
and then {x) — n{a 0 , a 1} ..., a^^x, 1) M_1 . 
We shall see later that cf>{x) can always be expressed as the 
product of n factors in the form 
</> (x) = a 0 (x - «])(« - a 2 ) ... (x — On), 
where the a’s are real or complex numbers. Then 
4>'(x) = a 0 X(x — a 2 )(x -a 3 )...(x- a n ),