Once you've learned that the derivative of f(x)=x² is 2x, it's natural to wonder — what happens when you differentiate other functions? Surprisingly, every function ends up with a completely different-looking derivative. Differentiate sin x and you get cos x; differentiate cos x and you get −sin x. Differentiate eˣ (the natural constant e raised to the x) and you get eˣ, unchanged; differentiate ln x (the natural logarithm) and you get 1/x.
Even though these formulas all look completely different, they actually all come from the exact same principle. No matter the function, the definition of a derivative is always the same: f'(a) = lim(h→0) [f(a+h)−f(a)]÷h. It's only the shape of the function that changes what result you actually get when you compute this limit. So while memorizing the formulas matters, understanding that "this formula is really just the slope of a secant line converging to a single value through a limit" makes them stick far more solidly.
eˣ in particular is an extraordinary function. It's the one function family that stays exactly the same shape no matter how many times you differentiate it — the only function "invariant under differentiation" (more precisely, this includes any constant multiple of eˣ too). Because of this property, eˣ naturally shows up whenever you need to describe something whose "rate of change is proportional to its current value" — population growth, radioactive decay, compound interest — because it means the changing quantity itself equals its own rate of change.
Why the derivative of sin x is cos x becomes easy to see once you look at the graphs. When the sin curve is at its highest point (its maximum), it's momentarily flat, so the tangent slope is 0. Plug that same x-value into the cosine function, and you get exactly cos(π/2)=0! Wherever the sin curve is rising steeply, the cosine value is large and positive; wherever it's decreasing, the cosine value is negative. Lay the two graphs side by side, and you can watch "the flow of the slope" get carried straight over into the other graph.
When studying this with kids, rather than having them memorize the formulas outright, it works better to first have them spot, by eye, where the tangent line goes flat (slope 0) and where it's steepest (maximum slope) on the graph, then pair those points up with where they land on the derivative graph. On our activity page, you can pick any function among sin, cos, eˣ, and ln x, compare the tangent line and derivative graph side by side, and directly compute the limit with a very small h to verify that the formula checks out exactly.