Assume that I have the following expression:
$$f(x,c) = \int_{0}^{x}g(\mathbf{\Psi}(x)^\intercal\mathbf{c}) dx$$
where $f(x,c)$ is a scalar, $x$ is a scalar, $\mathbf{\Psi}(x)$ is a $N$-by-$1$ vector, $\mathbf{c}$ is a $N$-by-$1$ vector, and $g:\mathbb{R}\rightarrow\mathbb{R}$ is a scalar-valued function. I'd like to know the partial derivatives of this expression with respect to $\mathbf{c}$:
$$\nabla_\mathbf{c}f(x,c) = ?$$
where $\nabla_\mathbf{c}=[\partial/\partial c_{1},...,\partial/\partial c_{N}]^\intercal$. Do you know how I could solve this? I would appreciate any full or partial solutions!