[Math] Why do you change the index of a power series when you differentiate it

Question

Why do you change the index of a power series when you differentiate it?

$$\sum_{n=0}^\infty \frac{d}{dx}(-x)^n=\sum_{n=1}^\infty n(-x)^{n-1}(-1)$$

A slow, dumbed-down explanation would be appreciated.

Answer 1

Notice what happens if we don't change the index, we end up with

$$\frac d{dx}x^n=nx^{n-1}$$

But when we sum it up from $n=0$ to $\infty$, the $n=0$ case ends up as $0$. And so it is removed from the sum. Also, you might want to check your result, it doesn't seem quite right.