Page:Calculus Made Easy.pdf/164

 of $$x$$, which is unchanged by differentiation? Accordingly; let us assume as a general expression that

(in which the coefficients $$A$$, $$B$$, $$C$$, etc. will have to be determined), and differentiate it.

Now, if this new expression is really to be the same as that from which it was derived, it is clear that $$A$$ must $$=B$$; that $$C=\dfrac{B}{2}=\dfrac{A}{1\cdot 2}$$; that $$D = \dfrac{C}{3} = \dfrac{A}{1 \cdot 2 \cdot  3}$$; that $$E = \dfrac{D}{4} = \dfrac{A}{1 \cdot 2 \cdot 3 \cdot 4}$$, etc.

The law of change is therefore that

If, now, we take $$A=1$$ for the sake of further simplicity, we have

Differentiating it any number of times will give always the same series over again.

If, now, we take the particular case of $$A=1$$, and evaluate the series, we shall get simply