The Function That Is Its Own Derivative
The number \(e \approx 2.718\) is special because of how it grows. The derivative of \(e^x\) is simply \(e^x\):
\[\frac{d}{dx}(e^x) = e^x\]
This means the height of the graph is always equal to its slope.
The Derivative of the Natural Log
The inverse of growth is the logarithm. The derivative of \(\ln(x)\) is:
\[\frac{d}{dx}(\ln x) = \frac{1}{x}\]
Worked Examples
Example 1: Scaled Growth
Find the derivative of \(f(x) = 5e^x\).
- Using the Constant Multiple Rule: \(5 \cdot \frac{d}{dx}(e^x) = 5e^x\).
- Result: \(f'(x) = 5e^x\).
Example 2: Combination
Find the derivative of \(f(x) = e^x + \ln x + x^2\).
- Differentiate term by term: \(e^x + \frac{1}{x} + 2x\).
- Result: \(f'(x) = e^x + \frac{1}{x} + 2x\).
The Bridge to Quantum Mechanics
In Quantum Mechanics, the probability of finding a particle in a certain state often decays exponentially. For instance, the wavefunction for a particle tunneling through a barrier is proportional to \(e^{-kx}\). The derivative of this function (which we'll handle with the Chain Rule soon) tells us the rate of decay, or how quickly the particle's presence "fades" into the forbidden region.