The Power Rule
Using the limit definition every time is slow. Mathematicians found a pattern for functions of the form \(f(x) = x^n\). The derivative is:
\[\frac{d}{dx}(x^n) = nx^{n-1}\]
Simply bring the power down in front and subtract one from the exponent.
Worked Examples
Example 1: Basic Application
Find the derivative of \(f(x) = x^5\).
- Bring down the 5: \(5x^{...}\)
- Subtract 1 from the exponent: \(5x^{5-1} = 5x^4\)
- Result: \(f'(x) = 5x^4\).
Example 2: Square Roots
Find the derivative of \(f(x) = \sqrt{x}\).
- Rewrite as a power: \(f(x) = x^{1/2}\)
- Bring down the 1/2: \(\frac{1}{2}x^{...}\)
- Subtract 1: \(\frac{1}{2}x^{1/2 - 1} = \frac{1}{2}x^{-1/2}\)
- Rewrite: \(\frac{1}{2\sqrt{x}}\)
- Result: \(f'(x) = \frac{1}{2\sqrt{x}}\).
Example 3: Negative Exponents
Find the derivative of \(f(x) = \frac{1}{x^3}\).
- Rewrite: \(f(x) = x^{-3}\)
- Power rule: \(-3x^{-4}\)
- Rewrite: \(-\frac{3}{x^4}\)
- Result: \(f'(x) = -\frac{3}{x^4}\).
The Bridge to Quantum Mechanics
Many potential energy wells in Quantum Mechanics are modeled as powers of \(x\). For example, the Harmonic Oscillator has a potential \(V(x) = \frac{1}{2}kx^2\). To find the force, we take the derivative: \(F = -\frac{dV}{dx} = -kx\). The Power Rule is our primary tool for converting energy landscapes into forces.