Differential Equation derivative question I was given this example in class:
$$\frac{dy}{dt}+\frac{y}{2}=\frac{1}{2}e^\frac{t}{3}$$
Finding $$\mu=e^\frac{t}{2}$$
After multiplying the whole equation we get:
$$\frac{dy}{dt}\cdot{e^\frac{t}{2}}+\frac{y}{2}\cdot{e^\frac{t}{2}}=\frac{1}{2}e^\frac{t}{3}\cdot{e^\frac{t}{2}}$$
However, after applying some algebra she does the following:

$$e^\frac{t}{2}\cdot{\frac{dy}{dt}}\longrightarrow{}\frac{d}{dt}(e^\frac{t}{2}\cdot{y})$$

How does that work and when is it valid?
Thanks!

$\frac{dy}{dt}\cdot{e^\frac{t}{2}}+\frac{y}{2}\cdot{e^\frac{t}{2}}=\frac{d}{dt}(e^{\frac{t}{2}}\cdot y)$ by the law for differentiating a product.
