    \begin{align*}
    |f_{c, x'', y''}'(x)| &= c \cdot \frac{f_{c, x'', y''}(x)}{x} = c \cdot y'' x^{-c-1} (x'')^c\\
    &\ge c \cdot \frac{y'\left(\frac{x+x'-x''}{x'}\right)^{-c}}{x+x'-x''} = c \cdot \frac{f_{c, x_0, y_0}(x+x'-x'')}{x+x'-x''}\\
    &= |f_{c, x_0, y_0}'(x+x'-x'')|.
    \end{align*}
