- #1
Screwdriver
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Homework Statement
Say that [tex]f(x)[/tex] is some function whose second derivative exists and say [tex]u(x, t)=f(x + ct)[/tex] for [tex]c > 0[/tex]. Determine
[tex]\frac{\partial u}{\partial x}[/tex]
In terms of [tex]f[/tex] and its derivatives.
Homework Equations
PD Chain rule.
The Attempt at a Solution
Say that x and y are both functions of f, ie. [tex]x = x(f)[/tex] and [tex]y = y(f)[/tex]. Then,
[tex]\frac{\partial u}{\partial x}=\frac{\partial }{\partial x}u(x(f),y(f))\frac{dx}{d f}(x(f))[/tex]
[tex]\frac{\partial u}{\partial x}=\frac{\partial }{\partial x}u(x,y)\frac{dx}{d f}(x(f))=\frac{\partial }{\partial x}f(x+ct)\frac{dx}{d f}(x+ct)[/tex]
[tex]\frac{\partial u}{\partial x}=f'(x+ct)[/tex]
Does that make any sense? It's mainly the notation that I don't understand, especially the "let x and y be functions of the same variable part."