Why e^x is Special: It Equals Its Own Derivative

Explore how different exponential bases relate to their derivatives
Base (a): 2.718
f(x) = a^x
f'(x) = ln(a) · a^x
Tangent line

Current Values at x = 0

Function value
1.00
Derivative value
1.00
Ratio (derivative/function)
1.00

Key Insight:

When a = e: The blue curve (e^x) and red curve (derivative) perfectly overlap! This means e^x equals its own derivative at every point.

For other bases: The derivative is scaled by ln(a). When a < e, the derivative is smaller than the function. When a > e, the derivative is larger.

Why this matters: This unique property makes e the natural choice for modeling growth processes where the rate of change is proportional to the current amount.