Derivatives of Exponential and Logarithmic Functions

Most functions change shape when you differentiate them — a parabola flattens to a line, a cubic drops to a parabola. One function refuses. There is a special number, e \approx 2.718, for which the exponential e^{x} is exactly its own derivative:

\frac{d}{dx} e^{x} = e^{x}

Its slope at every point equals its own height. That self-replicating property is what makes e the natural base for growth. For any other positive base a a constant tags along:

\frac{d}{dx} a^{x} = a^{x}\,\ln a

When a = e the factor \ln e = 1 disappears, and you are back to e^{x} reproducing itself.

The natural logarithm

The natural logarithm \ln x is the inverse of e^{x} — it undoes the exponential. Its derivative is one of the cleanest results in all of calculus:

\frac{d}{dx}\ln x = \frac{1}{x}

\dfrac{d}{dx} e^{x} = e^{x} — the exponential is the unique fixed point of differentiation;
\dfrac{d}{dx} a^{x} = a^{x}\,\ln a for any base a > 0;
\dfrac{d}{dx}\ln x = \dfrac{1}{x};
with the chain rule, \dfrac{d}{dx} e^{kx} = k\,e^{kx}.

See it being its own slope

Here is y = e^{x}. Because \frac{d}{dx}e^{x} = e^{x}, the height of the curve at any point is also its steepness there — where the curve is tall it is also rising fast.