Derivative of e^x
e^x is the only function (up to a constant multiple) that is its own derivative — the property that defines the number e in the first place.
Step-by-step solution
You are differentiating f(x) = ex, the exponential with base e ≈ 2.71828 and exponent exactly x.
For a general base, d/dx[ax] = ax ln(a). With a = e, the factor ln(e) = 1 disappears.
f′(x) = ex — identical to the original function.
Why it works
This self-reproducing property isn't a coincidence about e; it's e's job description. Among all exponential curves a^x, exactly one has slope 1 at x = 0, and we name its base e. Every other base grows proportionally to itself too, but with a constant of proportionality ln(a) — only base e makes that constant equal to 1.
Because differentiating e^x changes nothing, every higher derivative is also e^x, and the function is the backbone of solutions to growth and decay equations: any process whose rate of change is proportional to its current size (compound interest, population growth, radioactive decay with a sign flip) is an e^x in disguise.
Positive bases and real domains
Exp(x) is defined and differentiable for every real x. At x = 0 the point is (0, 1), its slope is 1, and its tangent line is y = 1 + x. All its higher derivatives equal exp(x).
The general real exponential rule aˣ ln(a) assumes a positive constant base a. For a = 1 it gives zero, since 1ˣ is constant. Negative bases do not define a real exponential function for every real x. The practice expression exp(x) + xᵉ is considered on x > 0 for the usual real-power definition.
Common mistakes
- Applying the power rule and writing xex−1. The power rule is for a variable base and constant exponent (xⁿ); here the base is constant and the exponent varies — a completely different rule.
- Forgetting the chain rule on e raised to anything other than plain x: d/dx[e^(x²)] = 2xe^(x²).
- Treating e like a variable and trying to differentiate "with respect to e."
Practice problems
Differentiate 5ex
Answer: 5ex.
Differentiate ex + xe
Answer: ex + e·xe−1 — the second term uses the power rule since e is constant.
Find the slope of ex at x = 0
Answer: e⁰ = 1 — the defining property of e.