what does "derivative" means in image processing?
Look at the numbers in the filter kernel in just 1 dimension for simplicity. For a Sobel and  Prewitt matrix you have something that roughly looks like this 

[-1,0,1]. 

Convolving this with your image basically computes the difference between the pixel values of the neighboring pixels. You apply 0 to the current pixel, 1 to the pixel on the right and -1 to the pixel on the left. This gives a first order difference: 

next pixel - previous pixel, 

i.e. first derivative.

But now look at a Laplacian operator. It looks something like [1, -2, 1]. This computes the difference of differences. To see how, note that

[1,-2,1] corresponds to

next - 2 x current + previous

i.e.

next - current - current + previous

i.e.

(next-current) - (current-previous)

Now notice how this is a diference of differences. (next - current) is like a 1st derivative. (current - previous) is like 1st derivative. Their difference is like a 2nd derivative.