Given a moving mesh $0=x_0(t)<x_1(t)<\cdots<x_N(t)<x_{N+1}(t)=1,$ where $t$ denotes the current time so that the mesh is moving with time. The linear basis function is then defined as $\phi_j(x,t)=\dfrac{x-x_{j-1}(t)}{x_j(t)-x_{j-1}(t)},$ for $x\in[x_{j-1}(t),x_j(t)]$ and $\phi_j(x,t)=\dfrac{x_{j+1}(t)-x}{x_{j+1}(t)-x_j(t)},$ for $x\in[x_j(t),x_{j+1}(t)]$ and $0$ otherwise, for all $j=1,\cdots,N.$ Then it comes with the conclusion that a direct calculation shows that $$ \dfrac{\partial\phi_j}{\partial t}(x,t)=-\dfrac{\partial\phi_j}{\partial x}(x,t)X_t(x,t), $$ where $X_t(x,t)$ is the linear interpolant of the nodal mesh speeds, i.e, $$X_t(x,t)=\sum\limits_{j=1}^N\dfrac{dx_j}{dt}(t)\phi_j(x,t).$$
However, I am not able to deduce this equation. Is anybody able to do that?