Why does taking the derivative with respect to time of an integral (with respect to tau) result in a function of time?

Question

I am having trouble understanding why line 3 to line 4 works, specifically when the derivative of u with respect to time is taken.

u is an integral of a function of tau with respect to tau, but taking the derivative with respect to time results in f(t).

How is this possible?

Answer 1

For any continuous function $f$, we have that

$$\frac{d}{dx} \left(\int_a^x f(y)dy \right) = f$$

this is called the Fundamental theorem of calculus.

In your case, you have $t$ in the upper bound of the function. for every $t$, $\int_0^t f(\tau)d\tau$ is some number, so you have a function of $t$ (not of $\tau$.