Nested Functions
The Chain Rule is the most important tool in calculus. It tells us how to differentiate a function inside another function: \(f(g(x))\).
\[\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)\]
Think of it as: "Derivative of the outside (leaving the inside alone), times the derivative of the inside."
Worked Examples
Example 1: Power of a Function
Find the derivative of \(f(x) = (3x^2 + 1)^4\).
- Outside: \(u^4 \to 4u^3\).
- Inside: \(3x^2 + 1 \to 6x\).
- Combine: \(4(3x^2 + 1)^3 \cdot 6x = 24x(3x^2 + 1)^3\).
- Result: \(f'(x) = 24x(3x^2 + 1)^3\).
Example 2: Trig with Argument
Find the derivative of \(f(x) = \sin(5x)\).
- Outside: \(\sin(u) \to \cos(u)\).
- Inside: \(5x \to 5\).
- Combine: \(\cos(5x) \cdot 5 = 5\cos(5x)\).
- Result: \(f'(x) = 5\cos(5x)\).
Example 3: Exponential with Function
Find the derivative of \(f(x) = e^{x^2}\).
- Outside: \(e^u \to e^u\).
- Inside: \(x^2 \to 2x\).
- Combine: \(e^{x^2} \cdot 2x\).
- Result: \(f'(x) = 2xe^{x^2}\).
The Bridge to Quantum Mechanics
The Chain Rule is everywhere in Quantum Physics because waves are functions of "Phase." A typical wavefunction looks like \(\psi(x) = e^{ikx}\). When we apply the momentum operator \(\hat{p} = -i\hbar \frac{d}{dx}\), we use the Chain Rule: \(\frac{d}{dx} e^{ikx} = ik e^{ikx}\). This multiplication by \(ik\) is what extracts the momentum value from the wavefunction. Without the Chain Rule, we couldn't "read" the physical information stored in the quantum state.