# Gradient of multiple integral function

given the vector $\mathbf{x}=(x_{h})_{h=1}^{H}$, some known scalar functions $f(\cdot)$, $g(\cdot)$, $y(\cdot)$, and a known scalar $\eta$, let us define the following vector valued function:

$\Phi(\mathbf{x}) = \int_{-\infty}^{g(x_{H})} f_{H} \big( t_{H},y(x_{H}) \big) \int_{\eta - g(x_{H})}^{g(x_{H-1})} f_{H-1} \big( t_{H-1},y(x_{H-1}) \big) \\ \ldots \int_{\eta - \sum_{h=2}^{H} g(x_{h})}^{g(x_{1})} f_{1} \big( t_{1},y(x_{1}) \big) dt_{1} \ldots dt_{H-1} dt_{H}$.

Is there anything I can do to calculate the Gradient $\nabla_{\mathbf{x}} \Phi(\mathbf{x})$? Or, at least, to simplify the function $\Phi(\mathbf{x})$?

Thanks

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