Reversing the Derivative

Calculus 1 · Axiom Academy

Differentiation has a backwards. Run it the other way and you've found an antiderivative. What if we ran differentiation backwards? All term, the question was one-directional: hand over a function, get its derivative. The slope machine only ran forward. But almost every question worth asking in the second half of calculus runs it the other way — you know the rate , and you want the thing it's the rate of . Velocity back to position. A rate of growth back to a total. To do that, you need to undo a derivative. Press play and watch the slope machine run in reverse on f(x) = 2x . The forward direction multiplies by the power and drops it by one; reversing means doing the opposite — bump the power up by one, then divide by the new power. The animation rebuilds the function whose derivative is 2x . Add one to the power, divide by the new power. That single move, run in reverse, is the power rule for antiderivatives: (for ). Here's the catch the forward machine hides. Constants vanish when you differentiate — , and so does . So when you reverse, you can't know which constant was there. Drag C and watch: every parabola in the stack is a different antiderivative of 2x , yet they all have the identical slope at every x . That whole family is the answer — written x^2 + C . Slide C as far as you like — the slope readout never changes. Differentiation throws the constant away, so an antiderivative is never one curve; it's a whole parallel family.

This is the written version of the interactive lesson above. See the full Calculus 1 course.