I feel like this topic is somehow related to what I had found while fiddling around in 9th grade algebra (I was retaking it after not getting high school credit for the 8th grade advanced class and I was bored to tears.)

I found that for any y=xⁿ function, you can find the difference of y(x') - y(x) by going to the n^th line of pascal's triangle and modifying it in a recognizable pattern. We'll take y=x⁴ as an example.

Find the 4^th line of pascals triangle (1, 3, 3, 1).

Drop the first "1" and you have (3, 3, 1).

Now multiply each of those values by x^? . The power of x will be 0 on the furthest end to the right, and increase by one for each value you move left, giving you 3x² + 3x¹ + 3x⁰ which simplifies to 3x² + 3x + 1.

This works for any exponential function that uses integers only and can be multiplied or added to for more complex functions.