# Taylor series

#### dwsmith

##### Well-known member
This is in relation to multi-scale perturbation techniques.

$x''+\epsilon f(x,x') + x = 0$

$f(x,x') = (x^2-1)x'$

\begin{alignat*}{3}
f\left(x_0 + \epsilon x_1 + \cdots , \left(\frac{\partial}{\partial t} + \epsilon\frac{\partial}{\partial T}\right)(x_0 + \epsilon x_1 + \cdots)\right) & = & f(x_0 + \epsilon x_1 + \cdots , x_{0t} + \epsilon(x_{0T} + x_{1t}) + \cdots)\\
& = & f(x_0,x_{0t}) + \epsilon x_1f_x(x_0,x_{0T}) + \epsilon (x_{0T} + x_{1t})f_{x'}(x_0,x_{0T})\\
& & + \cdots
\end{alignat*}

In line two, why/how does $x_1$ move outside of $f(,)$? I can't remember why terms are what they are after the order 1 term. Could someone explain it to me?