Taylor Expanding Two Equations: Analysis & Results

[latentcorpse]

I have two equations:

\ddot{x}^\mu + \ddot{y}^\mu + \Gamma^\mu{}_{\nu \lambda} (x+y)(\dot{x}^\nu+\dot{y}^\nu)(\dot{x}^\lambda+\dot{y}^\lambda)=0
and
\ddot{x}^\mu + \Gamma^\mu{}_{\nu\lambda}(x) \dot{x}^\nu \dot{x}^\lambda=0

apparently if i taylor expand the first equation to first order and then subtract the second equation i should get

\ddot{y}^\mu + \frac{\partial \Gamma^\mu{}_{\nu\lambda}}{\partial x^\rho} \dot{x}^\nu \dot{x}^\lambda y^\rho = 0

i cannot show this. how do we go about taylor expanding something like that?

 

[Fredrik]

Same way you'd expand any other function f:\mathbb R^4\rightarrow \mathbb R.

f(x+y)=f(x)+y^\rho f_{,\rho}(x)+\mathcal O(y^2)