The Limit Definition
To find the slope at a single point, we take two points very close together and then let the distance between them shrink to zero. This is the formal definition of the derivative:
\[f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}\]
This is called the difference quotient.
Worked Examples
Example 1: Derivative of a Constant
Find the derivative of \(f(x) = 5\).
- \(f(x+h) = 5\)
- \(f(x+h) - f(x) = 5 - 5 = 0\)
- \(\frac{0}{h} = 0\)
- \(\lim_{h \to 0} 0 = 0\)
- Result: \(f'(x) = 0\). (Makes sense: a flat line doesn't change).
Example 2: Derivative of \(f(x) = x\)
Find the derivative of \(f(x) = x\).
- \(f(x+h) = x+h\)
- \(f(x+h) - f(x) = (x+h) - x = h\)
- \(\frac{h}{h} = 1\)
- \(\lim_{h \to 0} 1 = 1\)
- Result: \(f'(x) = 1\).
Example 3: Derivative of \(f(x) = x^2\)
Find the derivative of \(f(x) = x^2\).
- \(f(x+h) = (x+h)^2 = x^2 + 2xh + h^2\)
- \(f(x+h) - f(x) = (x^2 + 2xh + h^2) - x^2 = 2xh + h^2\)
- \(\frac{2xh + h^2}{h} = 2x + h\)
- \(\lim_{h \to 0} (2x + h) = 2x\)
- Result: \(f'(x) = 2x\).
The Bridge to Quantum Mechanics
When we define momentum in Quantum Mechanics, we are looking at how the wavefunction changes over a tiny distance. The limit definition ensures that we are looking at the local properties of the particle. The "h" in our limit is a mathematical abstraction, but in the physical world, Planck's constant \(h\) sets a fundamental limit on how small "tiny" can actually be.