# Why is it possible to do this

How is it possible that one can break up the derivative operator such as this:

$$\frac{dv}{dt}=t^2$$, then integrate like this,
$$\int^v_{v_{0}}dv = \int^t_0 t^2 dt$$, where$$v=v_{o}$$ when $$t=0$$. Especially in light of what most calculus teachers tell you; that the derivative symbol is not a fraction and should not be interpreted as a faction?

## Answers and Replies

The derivative is a limit of a fraction and the integral is a limit of a summation of function values times a step length.

dv/dt is a function v'(t), and since it is a derivative, it is integrable by the fundamental theorem of calculus to v(t) + C where C is an undetermined constant.
If f(t) = g(t) and f(t) is integrable, then $\int f = (\int g) + C$ where C is an undetermined constant. Your use of boundary conditions (v(0) = v0) allows you to determine the constant.
That's all that's being done here. The Leibnitz notation is just a neat way of writing it out, but don't take it too seriously until you learn the proper way to manipulate differential forms.

wow thanks slider. I never have thought about it that way, but now that you have explained it, it is very interesting!

This was discussed in another thread.

What you're really integrating on the right side is $$\int^v_{v_{0}} \frac {dv}{dt} dt$$