I know that \frac{d}{dx} \sin x = \cos x, but I am not sure why. Can you explain *why* this formula is true?

The best way, in math, to see *why* something is true is to prove it. So that’s what we’ll do.

So we want to show that \frac{d}{dx} \sin x = \cos x. Since \sin is such a basic function there’s not a lot of manipulation that we can do to get to \cos. Therefore, we’ll probably have to go back to the definition of the derivative, using limits, to prove this.

Just to save space, I won’t go through it here, but the whole argument is clearly laid out here. Check it out and let me know if you have any questions!