First variation of functional

Dear all,

I am trying to find governing equations of Third-Order Beam Theory from its displacement field. For this, I need to find first variation of strains. I have written the following script:

clear all
close all
clc

syms z
syms u_0(x, t) phi(x, t) w_0(x, t)
syms h

c_1 = 4 / (3 * h^2);
u = u_0 + z * phi - c_1 * z^3 * (phi + diff(w_0, x, 1))
v = 0
w = w_0

strain_xx = diff(u, x, 1)
strain_xz = ( diff(u, z, 1) + diff(w, x, 1) ) / 2

Now I want to find first variation of strain_xx with respect to u_0, phi and w_0.

MATLAB have a functionalDerivative function, but I guess it is not exactly the first variation. Is there any built-in function which gives first variation of a functional.

Thank you.