Derivative of 2^x
Differentiating 2^x reveals the hidden tax every exponential pays for not using base e: a factor of ln 2.
Step-by-step solution
Any exponential can be re-expressed: 2x = ex ln 2, because e raised to ln 2 is 2.
The exponent is u = x ln 2 with derivative ln 2 (a constant). So the derivative is ex ln 2 · ln 2.
Substituting ex ln 2 = 2x gives f′(x) = 2x ln 2.
Why it works
ln 2 ≈ 0.693, so 2^x grows at only about 69% of its own height per unit step, while e^x grows at exactly 100% of its height. The constant ln(a) measures how far base a is from the "natural" growth rate — bases bigger than e get a factor above 1, bases smaller than e get one below 1.
This result matters far beyond the classroom: 2^x governs anything that doubles — computer memory, binary tree sizes, Moore's-law-style scaling — and its derivative tells you the instantaneous growth rate of a doubling process. The rule generalizes verbatim to any positive base: d/dx[a^x] = a^x ln a.
Common mistakes
- Applying the power rule to get x·2^(x−1). The base is constant and the exponent varies — power rule is exactly backwards here.
- Dropping the ln 2 and answering 2^x, as if every exponential were base e.
- Writing log₂ instead of ln — the factor is the natural log of the base, not a base-2 log.
Practice problems
Differentiate 10x
Answer: 10x ln 10.
Differentiate 2x/ln 2
Answer: 2x — the constant cancels.
Differentiate 2^(3x)
Answer: 3·2^(3x) ln 2 — chain rule adds the inner factor 3.