I have the following system of equations (for an ideal gas, from Exercise 2.1 from Fundamentals of Computational Fluid Dynamics): $$ \rho_t + \gamma p u_x + up_x = 0 $$ $$ p = (\gamma - 1) \left( e - \frac{1}{2} \rho u^2 \right) $$
I want to show that they are equivalent to a 1D-version of the Energy Density Conservation within the Compressible Euler Equations, which is as follows: $$ e_t + u e_x + \frac{p}{\rho} u_x = 0 $$
However, rearranging the second equation to the explicit $e = f(\gamma, \rho, u)$ form, and calculating the first partial derivatives yields: $$ e = \frac{p}{\gamma - 1} + \frac{1}{2} \rho u^2 $$ $$ e_t = \frac{p_t}{\gamma - 1} + \frac{1}{2} \left[ \rho u^2 \right]_t = \frac{p_t}{\gamma - 1} + \frac{u^2}{2} \rho_t + \rho u u_t $$ $$ e_x = \frac{p_x}{\gamma - 1} + \frac{1}{2} \left[ \rho u^2 \right]_x = \frac{p_x}{\gamma - 1} + \frac{u^2}{2} \rho_x + \rho u u_x $$ I am not able to see how these are equivalent: $$ \rho_t + \gamma p u_x + up_x \overset{?}{=} \left[ \frac{p_t}{\gamma - 1} + \frac{u^2}{2} \rho_t + \rho u u_t \right] + u \left[ \frac{p_x}{\gamma - 1} + \frac{u^2}{2} \rho_x + \rho u u_x \right] + \frac{p}{\rho} u_x$$