Under what condition(s) we can use the following relation: $\frac{df(x)}{du}=\frac{df(x)}{dx}\frac{dx}{du}$ Let $\frac{df(x)}{dx}$ the first derivative of function $f(x)$ and we assume that $x=x(u)$ (i.e. depends on $u$). Under what condition(s) we can use the following relation: $\frac{df(x)}{du}=\frac{df(x)}{dx}\frac{dx}{du}$.

This is called the chain rule its conditions are in your case:
$f$ is differentiable and $x$ has continous first derivative. 
