Derivative of e^(2x)
e^(2x) is the simplest possible chain-rule exponential — and the gateway drug to differentiating e raised to anything.
Step-by-step solution
The exponent isn't plain x; it's the inner function u = 2x. That means chain rule: differentiate the outside, then multiply by the derivative of the inside.
Outside: d/du[eu] = eu. Inside: d/dx[2x] = 2.
f′(x) = e2x · 2 = 2e2x.
Why it works
The pattern generalizes instantly: for any constant k, d/dx[ekx] = k·ekx. The exponential survives unchanged and the inner slope k pops out front. This one-liner is why e^(kx) solves the differential equation f′ = k·f, which models everything from bacterial growth (k > 0) to drug elimination in the bloodstream (k < 0).
It also explains half-life and doubling-time intuition: e^(2x) is e^x running at double speed, so at every point its slope is twice as steep relative to the same height. Compressing an exponential horizontally is indistinguishable from scaling its growth rate.
Common mistakes
- Answering e^(2x) with no factor of 2 — forgetting the chain rule entirely.
- Writing 2x·e^(2x−1), a hybrid of the power rule and the exponential rule that is valid for neither.
- Putting the 2 in the exponent of the answer, e.g. e^(4x). The inner derivative multiplies out front; it doesn't re-enter the exponent.
Practice problems
Differentiate e^(−3x)
Answer: −3e^(−3x) — the constant k can be negative.
Differentiate 4e^(2x)
Answer: 8e^(2x).
Differentiate e^(x²)
Answer: 2x·e^(x²) — the inner derivative is now a function, not a constant.