The Product Rule
If you have two functions multiplied together, you cannot just multiply their derivatives. Instead, we use the Product Rule:
\[\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)\]
In other words: "Derivative of the first times the second, plus the first times the derivative of the second."
Worked Examples
Example 1: Polynomial and Trig
Find the derivative of \(f(x) = x^2 \sin x\).
- Let \(f = x^2\) and \(g = \sin x\).
- \(f' = 2x\) and \(g' = \cos x\).
- Apply rule: \((2x)(\sin x) + (x^2)(\cos x)\).
- Result: \(f'(x) = 2x\sin x + x^2\cos x\).
Example 2: Exponential and Log
Find the derivative of \(f(x) = e^x \ln x\).
- Let \(f = e^x\) and \(g = \ln x\).
- \(f' = e^x\) and \(g' = 1/x\).
- Apply rule: \(e^x \ln x + e^x \frac{1}{x}\).
- Result: \(f'(x) = e^x(\ln x + \frac{1}{x})\).
The Bridge to Quantum Mechanics
In Quantum Mechanics, we often calculate Expectation Values, which involve integrating expressions like \(\psi^* x \psi\). When we calculate the time evolution of these values, we have to use the Product Rule to account for how both the wavefunction and the operator change over time. This leads directly to the Heisenberg Equation of Motion, the quantum version of Newton's second law.