Page:Elementary algebra (1896).djvu/490

594. Derived Functions. To find the value of $f(x+h)$, when $f(x)$ is a rational integral function of $x$.

Let $f(x)=p_{0}^{x^{n}}+{p_{1}}z{x-1})+\ldots p_{n-1}x+P_{n}$; then

Expanding each term and arranging the result in ascending powers of h, we have

n

This result is usually written in the form

F@ +N =f) +f @)+ pre) + Bret “+E and the functions $f(x),f''(x),f'''(x)$, are called the first, second, third, derived functions of f(x).

Examining the coefficients of h, we see that to obtain f'(x) from f(x) we multiply each term in f(x) by the index of x in that term, and then diminish the index by unity.

Similarly we obtain f(x), f'(x), .

595. Equal Roots. If the equation f(x)=0 has r roots equal to a, then the equation f'(x)=0 will have r-1 roots equal to a.

Let \phi(x) be the quotient when f(x) is divided by (x-a)r; then f(x)=(x-a)r \phi(x).

Write x + h in the place of x; thus

f(x+h)=(x-a+h)r \phi(x+h);