calculus - Ways to differentiate $(x)(x+y)$

I checked the differentiation of $(x)(x+y)$ using an online derivative tool which gives the result:

$\frac{d}{dx}\left(\left(x\right)\left(x+y\right)\right) = x+y+x\left(\frac{d}{dx}\left(y\right)+1\right)$

But using a different tool I found that derivate is:

$\frac{d}{dx}\left(\left(x\right)\left(x+y\right)\right) = 2x+y$

If there's no partial differentiation involved then does it mean there are different ways to interpret the given problem? i.e.

In first case, it is interpreted as $f(g(x), h(x,y)) = (x)(x+y)$ ?
and in second case it is $f(g(x), h(x)) = (x)(x+y)$ ?

I don't understand how the product rule is getting applied here and why $y$ is constant in second case?

Answer

In the first case, the online solver "thinks" $y$ is a function of $x$ and therefore only indicates the derivative, while in the second case the program treats $y$ as an independent variable. In both cases the product rule is being applied.

To be more explicit:

Let $f(x)=x$ and $g(x)=x+y$. Then x(x+y)=f(x)g(x). Then the product rule says that

$(f(x)(g(x))'=f'(x)g(x)+f(x)g'(x)$.

Note that $f'(x)=1$ and that if $y$ is and independent variable from $x$ then $g'(x)=1+0=1$. Substituting in the product rule we obtain

$(f(x)(g(x))'= 1(x+y)+x(1) = x+x+y=2x+y$.

However if $y$ depends of $x$, $g'(x)= 1 + d/dx (y)$. When substituting again on the product rule, you obtain the other result.