The basic commutation in evolution of Ricci flow

Question

Many references compute the evolution equation in Ricci flow like this, for example compute the evolution of Christoffel symbols $$\partial_{t} \Gamma^{k}_{ij} = {} \frac{1}{2}\partial_{t}(g^{kl}(\partial_{i} g_{jl} + \partial_{j} g_{il} - \partial_{l} g_{ij})) \\ = {} \frac{1}{2}(\partial_{t}g^{kl})(\partial_{i} g_{jl} + \partial_{j} g_{il} - \partial_{l} g_{ij}) + \frac{1}{2}g^{kl}(\partial_{t} \partial_{i} g_{jl} + \partial_{t} \partial_{j} g_{il} - \partial_{t} \partial_{l} g_{ij}) = \frac{1}{2}(\partial_{t}g^{kl})(\partial_{i} g_{jl} + \partial_{j} g_{il} - \partial_{l} g_{ij}) + \frac{1}{2}g^{kl}(\partial_{i} \partial_{t} g_{jl} + \partial_{j} \partial_{t} g_{il} - \partial_{l} \partial_{t} g_{ij})$$ I am puzzled about why the equation $\partial_{t} \partial_{l} g_{ij} = \partial_{l} \partial_{t} g_{ij}$ can hold? Since all common derivatives on natural frame can commute? I am not sure the real reasons.