Finding the second derivative of multvariable function

[newguy1234]

Hello I am trying to figure out the second derivative of

\frac{\partial^2 z}{\partial x^2} and \frac{\partial^2 z}{\partial t^2}

z(u) and u=x-vt

i found the first derivate of \frac{\partial z}{\partial x} to be
\frac{\partial z}{\partial x}= \frac{\partial z}{\partial u} * \frac{\partial u}{\partial x} = \frac{\partial z}{\partial u} *1

and

\frac{\partial z}{\partial t} = \frac{\partial z}{\partial u} * \frac{\partial u}{\partial t}= \frac{\partial z}{\partial u}*v

after this i am clueless on how to compute
(\frac{\partial }{\partial t})\frac{\partial z}{\partial u}*v
and
(\frac{\partial }{\partial x})\frac{\partial z}{\partial u}*1

i am new here am hoping to receive some help i will greatly appreciate it!

 

[samee]

Are you trying to say that z(u)=x-vt?

 

[newguy1234]

i guess the answer should be for \frac{\partial^2 z}{\partial t^2} = v^2 * \frac{\partial^2t }{\partial u^2}

im not sure at all how to get this tho any help?

 

[newguy1234]

i guess I am trying to say that z is a function of u and u is a function of x and t

 

[HallsofIvy]

Hello I am trying to figure out the second derivative of

\frac{\partial^2 z}{\partial x^2} and \frac{\partial^2 z}{\partial t^2}

z(u) and u=x-vt

i found the first derivate of \frac{\partial z}{\partial x} to be
\frac{\partial z}{\partial x}= \frac{\partial z}{\partial u} * \frac{\partial u}{\partial x} = \frac{\partial z}{\partial u} *1

and

\frac{\partial z}{\partial t} = \frac{\partial z}{\partial u} * \frac{\partial u}{\partial t}= \frac{\partial z}{\partial u}*v

No, since u= x- vt, \partial u/\partial t= -v, not v.

after this i am clueless on how to compute
(\frac{\partial }{\partial t})\frac{\partial z}{\partial u}*v
and
(\frac{\partial }{\partial x})\frac{\partial z}{\partial u}*1

i am new here am hoping to receive some help i will greatly appreciate it!

The "v" and "1" are constants. So you are just differentiating a function of x and t again:
\frac{\partial}{\partial x}\frac{\partial z}{\partial u}= \frac{\partial^2 z}{\partial u^2}\frac{\partial u}{\partial x}
and, of course, \partial u/\partial x is 1.

Likewise
\frac{\partial}{\partial t}\frac{\partial z}{\partial u}= \frac{\partial^2 z}{\partial u^2}\frac{\partial u}{\partial t}
and \partial u/\partial t is -v.