Differentiation is, for me, where understanding of what’s going on slips into a mess of tools, symbols and formulas. The grapher application helps greatly with this – after doing the necessary mental gymnastics, it’s useful to see the relationship between the derivation and the original. Here’s 
and its derivative 
. You can see the latter changing nicely with the former.
The material I’m revising introduces differentiation from its bare definition, calculating the gradient between two different points on a smooth function, then making those points arbitrarily close from the quotient:
Then there’s lots of nice reasoning to find results for different functions, with new techniques enabling further discoveries. For example, here’s the logic for the sine function:
If
and 
is more than zero, and 
, after collecting like terms we end up with:
it can be shown (using geometric and algebraic arguments) that:
and 
which ends up with:
leaving 