How does one compute a variational derivative? The expected regression loss is given as:$$E[L]=\int\int \{y(\mathbf x)-t\}^2 p(\mathbf x,t)d\mathbf xdt$$
To minimise the expected loss,Euler Lagrange equation is used which goes like this in the general form:$$ \frac{\delta F}{\delta y}=\frac{d}{dx}\left( \frac{\delta F}{\delta y'}\right)$$ where $F$ is the functional i.e function of function $\mathit y(x)$.
In our case $F$ would correspond to $E$ and $\mathit y$ to $L$.
Please show a step wise application of this equation to yield the following expression
$$\frac{\delta E[L]}{\delta \mathit y(\mathbf x)}=2 \int\{\mathit y(\mathbf x)-t\}p(\mathbf x,t) 
dt=0.$$
Also, please suggest a good reference for all the calculus used in Machine Learning.
 A: I will carry out the steps in a way I hope is clear enough to indicate what assumptions must be made about $y,$ $p,$ and $\delta$ to justify the steps.
When a function $y$ is a local minimum of a functional $\mathcal L,$ adding a sufficiently small multiple $h$ of a "test function" $\delta$ cannot decrease the value of the functional.  (Usually a test function is assumed to be arbitrarily smooth and of compact support; in more complex situations it may have to satisfy constraints imposed by boundary conditions.)  That is, writing
$$\mathcal{L}[y] = \iint \left(y(x)-t\right)^2\,p(x,t)\,\mathrm{d}x\mathrm{d}t,$$
it must be the case that for all $|h|$ in some small neighborhood $U$ of $0$ (which may depend on $\delta$),
$$\eqalign{
0 &\le \mathcal{L}[y+h\delta] - \mathcal{L}[y] \\
&= 2h\iint \delta(x) \left(y(x)-t\right)\,p(x,t)\,\mathrm{d}x\mathrm{d}t + h^2 \iint \delta^2(x) \,p(x,t)\,\mathrm{d}x\mathrm{d}t
.}\tag{1}$$
Given $\delta,$ the right hand side is a quadratic function of $h$ with a zero at $h=0.$ (This assertion follows from the fact that the integral of $\delta^2$ is finite, which guarantees both integrals in $(1)$ exist and are finite provided $y$ and $p$ are not too "badly behaved," which is implicitly assumed in the question.)
In order to be non-negative throughout $U$, this zero must be the vertex of the quadratic; that is,
$$0 = 2\iint \delta(x) \left(y(x)-t\right)\,p(x,t)\,\mathrm{d}x\mathrm{d}t$$
This says the function
$$x \to \int \left(y(x)-t\right)p(x,t)\,\mathrm{d}t$$
is orthogonal to all test functions $\delta,$ implying (by virtue of some assumed "not bad behavior" of $p$) it is almost everywhere zero, QED.

Generally, to find the derivative of a functional $\mathcal L,$ you proceed as in ordinary differential Calculus to form the difference quotient as in $(1)$
$$\frac{\mathcal{L}[y+h\delta] - \mathcal{L}[y]}{h} = \iint \delta(x)\, 2\left(y(x)-t\right)\,p(x,t)\,\mathrm{d}x\mathrm{d}t + o(h).$$
The right hand side applies the linear operator
$$D[\delta] = \int \delta(x)\,\left[2\int (y(x)-t)\,p(x,t)\,\mathrm{d}t\right]\,\mathrm{d}x$$
represented by the generalized function
$$x \to 2\int (y(x)-t)\,p(x,t)\,\mathrm{d}t$$
to $\delta$ and adds an error term $o(h)$ that is vanishingly small compared to $h.$  This conforms with the (usual) definition of the derivative, allowing us to write
$$\frac{\partial \mathcal{L}}{\partial y} = 2\int (y(x)-t)\,p(x,t)\,\mathrm{d}t.$$
A: The straightforward application of Euler Lagrange equation would lead to the sought result.
First, the derivative of the integrated expression with respect to $y'$ is zero, simply because $y'$ is nowhere to be seen. Here  $F(y(\mathbf x),\mathbf x)=\int (y(\mathbf x)-t)^2 p(\mathbf x,t)dt$, so we get: $\frac{\partial}{\partial y'}F=0$
Second, the derivative of $F$ with respect to $y(\mathbf x)$ is trivial, no need to even apply Leibnitz rule:
$$F_y=  \frac{\partial}{\partial y(\mathbf x)}\int (y(\mathbf x)-t)^2 p(\mathbf x,t)dt
=2\int (y(\mathbf x)-t) p(\mathbf x,t)dt$$
So, there you get the equation that you're looking for:
$$F_y=2\int (y(\mathbf x)-t) p(\mathbf x,t)dt=0$$
This condition minimizes the expectation.
As to reference to all math that's used in machine learning, there's no such a thing. Whatever the researches decides to use is a fair game. For instance, number theory and abstract algebra concepts can be used, like in this paper. The most common tools tend to be linear algebra, real analysis (calculus), probability and typical computer science toolset such as graph theory.
