Un-Differentiating
If differentiation takes a position and gives you velocity, what takes velocity and gives you back position? This process is called Antidifferentiation. An antiderivative of \(f(x)\) is a function \(F(x)\) such that \(F'(x) = f(x)\).
Worked Examples
Example 1: The Power Rule in Reverse
Find an antiderivative for \(f(x) = x^2\).
- We know \(\frac{d}{dx}(x^3) = 3x^2\).
- To get just \(x^2\), we need to divide by 3: \(\frac{x^3}{3}\).
- Result: \(F(x) = \frac{1}{3}x^3\).
Example 2: Trig Reverse
Find an antiderivative for \(f(x) = \cos x\).
- We know \(\frac{d}{dx}(\sin x) = \cos x\).
- Result: \(F(x) = \sin x\).
The Bridge to Quantum Mechanics
In Quantum Mechanics, we often know the Momentum of a particle and want to find its Wavefunction. Since the momentum operator is a derivative, finding the wavefunction requires finding the antiderivative. This "reverse" process is how we move from observable quantities (like energy and momentum) back to the underlying quantum state \(\psi\).