The Derivative of eˣ: The Function That Is Its Own Rate of Change

The Derivative of eˣ: The Function That Is Its Own Rate of Change | Ideasthesia

The exponential function eˣ has a remarkable property:

d/dx (eˣ) = eˣ

Its derivative is itself. The rate at which eˣ changes equals eˣ.

Here's the unlock: this isn't a coincidence — it's the definition of e. The number e ≈ 2.71828 is specifically the base where the exponential function equals its own derivative. Any other base gives a different constant factor.

eˣ is the fixed point of differentiation. It's the function that, when you ask "how fast is it changing?", answers "this fast" by pointing at itself.

Why This Is Remarkable

Most functions change at different rates than their values:

d/dx(x²) = 2x ≠ x²
d/dx(sin x) = cos x ≠ sin x
d/dx(ln x) = 1/x ≠ ln x

But eˣ is different. Its rate of change at any point equals its value at that point.

If eˣ = 10, it's growing at rate 10. If eˣ = 1000, it's growing at rate 1000.

The bigger it is, the faster it grows — proportionally. That's exponential growth at its purest.

The Definition of e

One way to define e:

e = lim[n→∞] (1 + 1/n)ⁿ ≈ 2.71828...

This is the limit of compound interest with continuous compounding: invest $1 at 100% interest, compounding n times per year, as n approaches infinity.

Alternatively, e is the unique number where:

lim[h→0] (eʰ - 1)/h = 1

This limit being exactly 1 is what makes d/dx(eˣ) = eˣ.

Proving d/dx(eˣ) = eˣ

From the definition of derivative:

d/dx(eˣ) = lim[h→0] (e^(x+h) - eˣ)/h = lim[h→0] (eˣ · eʰ - eˣ)/h = lim[h→0] eˣ(eʰ - 1)/h = eˣ · lim[h→0] (eʰ - 1)/h = eˣ · 1 = eˣ

The key step is that lim[h→0] (eʰ - 1)/h = 1 by the definition of e.

Derivatives of e^(something)

Using the chain rule:

d/dx(e^(f(x))) = e^(f(x)) · f'(x)

Examples:

d/dx(e^(2x)) = e^(2x) · 2 = 2e^(2x)

d/dx(e^(x²)) = e^(x²) · 2x = 2x·e^(x²)

d/dx(e^(sin x)) = e^(sin x) · cos x

The eˣ part differentiates to itself, then multiply by the chain rule factor.

Other Exponential Bases

For aˣ where a ≠ e:

d/dx(aˣ) = aˣ · ln(a)

The ln(a) factor appears because aˣ = e^(x·ln(a)).

d/dx(2ˣ) = 2ˣ · ln(2) ≈ 0.693 · 2ˣ

d/dx(10ˣ) = 10ˣ · ln(10) ≈ 2.303 · 10ˣ

Only for base e does the constant factor equal 1.

This is why mathematicians prefer e: it's the "natural" base where exponentials behave cleanly under differentiation.

The Differential Equation

The equation dy/dx = y is a differential equation. It asks: what function equals its own derivative?

The answer is y = Ceˣ for any constant C.

Check: d/dx(Ceˣ) = Ceˣ ✓

This is the general solution. The specific solution depends on an initial condition (the value of C).

Exponential Growth and Decay

Physical systems described by dy/dt = ky have solutions y = y₀e^(kt).

k > 0: Exponential growth (populations, compound interest, nuclear chain reactions)

k < 0: Exponential decay (radioactive decay, cooling, drug metabolism)

The differential equation dy/dt = ky says "rate of change proportional to current value." The solution is always exponential.

eˣ and Its Integral

Since d/dx(eˣ) = eˣ:

∫eˣ dx = eˣ + C

eˣ is also its own antiderivative. It's a fixed point under both differentiation and integration.

Taylor Series

eˣ has a beautiful Taylor series:

eˣ = 1 + x + x²/2! + x³/3! + x⁴/4! + ...

Differentiating term by term gives the same series — confirming that eˣ equals its own derivative.

Connection to Complex Numbers

Euler's formula: e^(ix) = cos(x) + i·sin(x)

This connects the exponential function to rotation. Differentiating:

d/dx(e^(ix)) = i·e^(ix) = i(cos x + i sin x) = -sin x + i cos x

The real part is -sin x = d/dx(cos x) ✓ The imaginary part is cos x = d/dx(sin x) ✓

Everything is consistent. The derivative of eˣ being eˣ underlies all of it.

Why e Is Special

e is the base that makes calculus clean:

d/dx(eˣ) = eˣ (no constant factor)
ln(e) = 1 (simplest logarithm)
d/dx(ln x) = 1/x (no constant factor)
The limit definition of derivative works out to exactly 1

Any other base introduces ln(a) factors everywhere. e is where those factors become 1.

The Core Insight

eˣ is the function that grows at a rate equal to its size.

This isn't just a mathematical curiosity — it's why e appears everywhere that growth is proportional to quantity. Populations grow this way. Money compounds this way. Radioactive decay follows this pattern.

The derivative d/dx(eˣ) = eˣ isn't a rule you memorize. It's the definition of what makes e special: the unique base where exponential growth is its own derivative.

When you see eˣ, you're seeing the purest form of proportional change.

Part 9 of the Calculus Derivatives series.