Taking My Time

Given any $x$,

$\cfrac { \partial x\,  }{ \partial  x } =1$

$\cfrac { \partial ^{ 2 }\, x }{ \partial \, x^{ 2 } } =0=\cfrac { \partial ^{ 2 }\, x }{ \partial \, x\partial t } \cfrac { \partial t\,  }{ \partial \, x } =\cfrac { \partial ^{ 2 }\, x }{ \partial t\partial t } \cfrac { \partial t\,  }{ \partial \, x } \cfrac { \partial t\,  }{ \partial \, x } =\cfrac { \partial ^{ 2 }\, x }{ \partial t^{ 2 } } \cfrac { \partial t\,  }{ \partial \, x } \cfrac { \partial t\,  }{ \partial \, x }$

So for all $x$,

$\cfrac { \partial ^{ 2 }\, x }{ \partial t^{ 2 } } \cfrac { \partial t\,  }{ \partial \, x } \cfrac { \partial t\,  }{ \partial \, x } =0$

ie,

$\cfrac{a}{v^2}=0$

In fact,

$\cfrac { \partial ^{ 2 }x\quad  }{ \partial \, x^{ 2 } } =0$

$=\cfrac { \partial \, \quad  }{ \partial \, x } \{ \cfrac { \partial x\,  }{ \partial \, x } \}$

$=\cfrac { \partial \, \quad  }{ \partial \, t } \{ \cfrac { \partial x\,  }{ \partial \, x } \} \cfrac { \partial t\,  }{ \partial \, x }$

$=\cfrac { \partial \, \quad  }{ \partial \, t } \{ \cfrac { \partial x\,  }{ \partial \, t } \cfrac { \partial t\,  }{ \partial \, x } \} \cfrac { \partial t\,  }{ \partial \, x }$

$=\cfrac { \partial \, \quad  }{ \partial \, t } \{ \cfrac { \partial x\,  }{ \partial \, t } \} \cfrac { \partial t\,  }{ \partial \, x } +\cfrac { \partial \, \quad  }{ \partial \, t } \{ \cfrac { \partial t\,  }{ \partial \, x } \} \cfrac { \partial x\,  }{ \partial \, t }$

$= \cfrac { \partial ^{ 2 }x\, \quad  }{ \partial \, t^{ 2 } } \cfrac { \partial t\,  }{ \partial \, x } -\cfrac { 1 }{ \{ \cfrac { \partial x\,  }{ \partial \, t } \} ^{ 2 } } \cfrac { \partial \, \quad  }{ \partial \, t } \{ \cfrac { \partial x\,  }{ \partial \, t } \} \cfrac { \partial x\,  }{ \partial \, t }$

$=\cfrac { \partial ^{ 2 }x\, \quad  }{ \partial \, t^{ 2 } } \cfrac { \partial t\,  }{ \partial \, x } -\cfrac { 1 }{ \{ \cfrac { \partial x\,  }{ \partial \, t } \}  } \cfrac { \partial ^{ 2 }x\, \quad  }{ \partial \, t^{ 2 } }$

$=\cfrac { \partial ^{ 2 }x\, \quad  }{ \partial \, t^{ 2 } } \cfrac { \partial t\,  }{ \partial \, x } -\cfrac { \partial ^{ 2 }x\, \quad  }{ \partial \, t^{ 2 } } \cfrac { \partial t }{ \partial x }=0$

non sequitur.