Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/265

 Evidently (B) is the value of (C) when $$t = 1$$. Considering (C) as a function of, we may write

which may then be expanded in powers of $$t$$ by Maclaurin's Theorem, (64), &sect; 145, giving

Let us now express the successive derivatives of $$F(t)$$ with respect to $$t$$ in terms of the partial derivatives of $$F(t)$$ with respect to $$x$$ and $$y$$. Let

then by (51), &sect;125,

But from (F),

and since $$F(t)$$ is a function of $$x$$ and $$y$$ through $$\alpha$$ and $$\beta$$,

or, since from (F), $$\tfrac{\partial\alpha}{\partial x} = 1$$ and $$\tfrac{\partial\beta}{\partial y} =1$$,

Substituting in (G) from (I) and (H),

Replacing $$F(t)$$ by $$F'(t)$$ in (J), we get

In the same way the third derivative is

and so on for higher derivatives.