Set up
r.v. $X \sim U(0, 1), Y \sim U(0, X), Z \sim U(X, 1)$.
Fact
j.p.d.f. $$ f(x, y, z) = \frac{1}{x} \frac{1}{1-x}. $$
What I did
j.c.d.f. is \begin{align} F(x, y, z) = \mathbb{P}(X \leq x, Y \leq y, Z \leq z) = x \times \frac{y}{x} \times \frac{z-x}{1-x} = \frac{y(z-x)}{1-x}. \end{align} Therefore \begin{align} f(x, y, z) = \frac{\partial^3}{\partial x \partial y \partial z} F(x, y, z) = \frac{1}{(1-x)^2} \leftarrow \mathrm{mismatch} \end{align}
Question
How can I ask for it? Or where am I going wrong?
Your "fact" is actually not factual.
You do not have the Joint Cumulative Distribution.
$\begin{align}\mathsf P(X\leqslant x, Y\leqslant y, Z\leqslant z)~ &=~\mathsf P(X\leqslant x)\,\mathsf P(Y\leqslant y,Z\leqslant z\mid X\leqslant x)\tag 1\end{align}$
Rather you begin with this:
$\begin{align}\hspace{9ex}x\times\dfrac yx\times\dfrac{z-x}{1-x} ~&=~ \mathsf P(X\leqslant x)\,\mathsf P(Y\leqslant y, Z\leqslant z\mid X=x)\tag 2\end{align}$
These are not the same thing at all.