I am given this result:
$$\frac{\partial}{\partial x(t)} \left[\lambda \int u(x(t)) f(t) \mathrm{d}t\right] = \lambda u^\prime(x(t)) f(t)$$ Where $\lambda$ is a constant, and we have the probability density f(t) of the variable $t$ and the functions $x(t)$ and $u(\cdot)$ of $t$ and $x(t)$ respectively.
How do I arrive to this result? Where am I mistaken? Using the chain rule I get:
$$\begin{align*}\frac{\partial}{\partial x(t)} \left[\lambda \int u(x(t)) f(t) \mathrm{d}t \right]&= \lambda \left[ \frac{\partial}{\partial x(t)} \left[\int u(x(t)) f(t) \mathrm{d}t \right]\right] \frac{\partial}{\partial x(t)} u(x(t)) \\ &= \lambda \underbrace{\left[ \frac{\partial}{\partial x(t)} \left[ \int u(x(t)) f(t) \mathrm{d}t \right]\right]}_{\text{this must be } f(t)?} u^\prime(x(t)) \end{align*}$$
So we must have: $$ \frac{\partial}{\partial x(t)} \left[ \int u(x(t)) f(t) \mathrm{d}t \right] = f(t) $$ But why does this hold?