Find the derivative of the solution of the equation $\ddot{x} =x + A\dot{x}^2$, with initial conditions $x(0) = 1$, $\dot{x}(0) = 0$, with respect to the parameter $A$ for $A = 0$.
1. **Solve the differential equation for \( A = 0 \):**
   The given differential equation is:
   \[
   \ddot{x} = x + A\dot{x}^2
   \]
   When \( A = 0 \), the equation simplifies to:
   \[
   \ddot{x} = x
   \]
   This is a standard second-order linear differential equation with constant coefficients. The characteristic equation is:
   \[
   r^2 - 1 = 0 \implies r = \pm 1
   \]
   Therefore, the general solution is:
   \[
   x(t) = C_1 e^t + C_2 e^{-t}
   \]
   Using the initial conditions \( x(0) = 1 \) and \( \dot{x}(0) = 0 \):
   \[
   x(0) = C_1 + C_2 = 1
   \]
   \[
   \dot{x}(0) = C_1 e^0 - C_2 e^0 = C_1 - C_2 = 0 \implies C_1 = C_2
   \]
   Substituting \( C_1 = C_2 \) into the first equation:
   \[
   2C_1 = 1 \implies C_1 = \frac{1}{2}, \quad C_2 = \frac{1}{2}
   \]
   Thus, the solution for \( A = 0 \) is:
   \[
   x(t) = \frac{1}{2} e^t + \frac{1}{2} e^{-t} = \cosh(t)
   \]

2. **Formulate the perturbed solution for small \( A \):**
   Assume the solution for small \( A \) is of the form:
   \[
   x(A, t) \approx f(t) + A g(t)
   \]
   where \( f(t) = \cosh(t) \) is the solution for \( A = 0 \).

3. **Substitute the perturbed solution into the original differential equation:**
   \[
   \ddot{x}(A, t) = \ddot{f}(t) + A \ddot{g}(t)
   \]
   \[
   \dot{x}(A, t) = \dot{f}(t) + A \dot{g}(t)
   \]
   \[
   \dot{x}(A, t)^2 = (\dot{f}(t) + A \dot{g}(t))^2 = \dot{f}(t)^2 + 2A \dot{f}(t) \dot{g}(t) + A^2 \dot{g}(t)^2
   \]
   Ignoring higher-order terms in \( A \), we get:
   \[
   \dot{x}(A, t)^2 \approx \dot{f}(t)^2 + 2A \dot{f}(t) \dot{g}(t)
   \]
   Substitute these into the original equation:
   \[
   \ddot{f}(t) + A \ddot{g}(t) = f(t) + A g(t) + A (\dot{f}(t)^2 + 2A \dot{f}(t) \dot{g}(t))
   \]
   Equate the coefficients of \( A \):
   \[
   \ddot{f}(t) = f(t)
   \]
   \[
   A \ddot{g}(t) = A g(t) + A \dot{f}(t)^2
   \]
   Simplifying, we get:
   \[
   \ddot{g}(t) = g(t) + \dot{f}(t)^2
   \]

4. **Solve for \( g(t) \):**
   We know \( f(t) = \cosh(t) \), so:
   \[
   \dot{f}(t) = \sinh(t)
   \]
   \[
   \dot{f}(t)^2 = \sinh^2(t)
   \]
   Thus, the equation for \( g(t) \) becomes:
   \[
   \ddot{g}(t) = g(t) + \sinh^2(t)
   \]
   The particular solution to this non-homogeneous differential equation can be found using the method of undetermined coefficients. The homogeneous part is:
   \[
   \ddot{g}(t) = g(t)
   \]
   The characteristic equation is:
   \[
   r^2 - 1 = 0 \implies r = \pm 1
   \]
   So the general solution to the homogeneous equation is:
   \[
   g_h(t) = C_1 e^t + C_2 e^{-t}
   \]
   For the particular solution, we use the fact that:
   \[
   \sinh^2(t) = \frac{1}{2} (\cosh(2t) - 1)
   \]
   Therefore, we look for a particular solution of the form:
   \[
   g_p(t) = A \cosh(2t) + B
   \]
   Substituting \( g_p(t) \) into the non-homogeneous equation:
   \[
   4A \cosh(2t) = A \cosh(2t) + \frac{1}{2} (\cosh(2t) - 1)
   \]
   Equating coefficients, we get:
   \[
   4A = A + \frac{1}{2} \implies A = \frac{1}{6}
   \]
   \[
   0 = B - \frac{1}{2} \implies B = \frac{1}{2}
   \]
   Thus, the particular solution is:
   \[
   g_p(t) = \frac{1}{6} \cosh(2t) + \frac{1}{2}
   \]
   Therefore, the general solution for \( g(t) \) is:
   \[
   g(t) = C_1 e^t + C_2 e^{-t} + \frac{1}{6} \cosh(2t) + \frac{1}{2}
   \]
   Using the initial conditions \( g(0) = 0 \) and \( \dot{g}(0) = 0 \):
   \[
   g(0) = C_1 + C_2 + \frac{1}{6} + \frac{1}{2} = 0 \implies C_1 + C_2 = -\frac{2}{3}
   \]
   \[
   \dot{g}(0) = C_1 - C_2 + 2 \cdot \frac{1}{6} \sinh(0) = 0 \implies C_1 = C_2
   \]
   Substituting \( C_1 = C_2 \) into the first equation:
   \[
   2C_1 = -\frac{2}{3} \implies C_1 = -\frac{1}{3}, \quad C_2 = -\frac{1}{3}
   \]
   Thus, the solution for \( g(t) \) is:
   \[
   g(t) = -\frac{1}{3} e^t - \frac{1}{3} e^{-t} + \frac{1}{6} \cosh(2t) + \frac{1}{2}
   \]

The final answer is \( \boxed{ g(t) = -\frac{1}{3} e^t - \frac{1}{3} e^{-t} + \frac{1}{6} \cosh(2t) + \frac{1}{2} } \)