Let $M$ be a smooth manifold. Let $f_t$ be a differentiable family of diffeomorphisms $M \to M$, $t \in (-\epsilon,\epsilon), f_0=id$. Let $X=\frac{df_t}{dt}|_{t=0}$. The Lie derivative of a metric $g$ is defined by
$$L_Xg=\frac{d}{dt}|_{t=0}f_t^*g$$ where $f^*_t$ denotes pullback by $f_t$. How can I show that in local coordinates, it can be expressed as $$(L_Xg)_{ij}=X^k\frac{\partial g_{ij}}{\partial x^k}+g_{ik}\frac{\partial X^k}{\partial x^j}+g_{jk}\frac{\partial X^k}{\partial x^i}$$ I tried to start with the expression $$(L_Xg)_{ij}=\frac{d}{dt}|_{t=0}g(f_t(x))(d_xf_t(\partial_i),d_xf_t(\partial_j))$$ but I can hardly simplify this expression. Can anyone show me how to simplify this? Thank you very much!