my question in the photo

47958

Looks like the question is incomplete. In this part:

"and \frac {\partial F(x+ct)}{\partial t} at (t=0).."

shouldn't that partial derivative be equal to something?

no I just want to show that the function is partial derivative and it will be total when t = 0

the question in an other way
If I have a function of x and t it's derivative with respect to t in case of t=0 is equal to it's dervative with respect to x why ? Why not ?

Consider the total derivates with respect to x and t:

\frac {dF}{dt}= \frac {\partial F}{\partial t} + \frac {\partial F}{\partial x}\frac {dx}{dt}

\frac {dF}{dx}= \frac {\partial F}{\partial x} + \frac {\partial F}{\partial t}\frac {dt}{dx}

If these are equal then

\frac {\partial F}{\partial t} + \frac {\partial F}{\partial x}\frac {dx}{dt} =\frac {\partial F}{\partial x} + \frac {\partial F}{\partial t}\frac {dt}{dx}


This could be true if \frac {dx}{dt} = \frac {dt}{dx} = 1