The answer has been given in the MO question Integral representation of higher order derivatives by P. Majer. The formula is that if $u$ is even, then the derivatives of the function $w$ defined by $w(x):=u(\sqrt x)$ satisfy $$w^{(k)}(x^2)=\frac{\\ (2x)^{-2k+1}}{(k-1)!\\ }\\ \int_0^x (x^2-t^2)^{k-1}u^{(2k)}(t) dt\\ $$

The answer has been given in the MO question Integral representation of higher order derivatives by P. Majer. The formula is that if $u$ is even, then the derivatives of the function $w$ defined by $w(x):=u(\sqrt x)$ satisfy $$w^{(k)}(x^2)=\frac{\\ (2x)^{-2k+1}}{(k-1)!\\ }\\ \int_0^x (x^2-t^2)^{k-1}u^{(2k)}(t) dt\\ $$