Why is $x\frac{dy}{dx} + y = \frac{d}{dx} (xy)$?

Question

In my physics textbook it is stated without further explanation that: $$x\frac{dy}{dx} + y = \frac{d}{dx} (xy)$$ How is this proven?

Answer 1

If $y=y(x)$ is a function of $x$, then this is just the product rule: if $f$ and $g$ are differentiable, then $fg$ is differentiable and $$\frac{d}{dx}(fg)=\frac{df}{dx}g+f\frac{dg}{dx}.$$

Answer 2

Take the product rule between $\frac{d}{dx}\left(xy\right)$

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

Answer 3

Use the product rule:

$$\frac{d(fg)}{dx}=f'g+g'f$$ $f=x, g=y$.

Hence $f'=1, g'=\frac{dy}{dx}$

Thus we get $\frac{d(fg)}{dx}=1\cdot y+x\cdot \frac{dy}{dx}=y+x\frac{dy}{dx}$