What is the derivative of the real function $f:\mathbb{R}\rightarrow \mathbb{R}$ defined by
$$f(x)=\int\_{D(x)}g(\mathbf{t},x)\ d\mathbf{t},$$

where $D(x)=\{\mathbf{t}=(t\_1,\ldots,t\_n)\in \mathbb{R}^n: t\_i \leq x,\ i=1,\ldots,n\}$ ?

I guess that under certain conditions the derivative is
$$f'(x)=\int\_{D(x)}\frac{d}{dx}g(\mathbf{t},x) \ d\mathbf{t}.$$

Am I correct ?