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 has an instantaneous growth rate of about 69% of its current value per input unit, 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.
Domain and instantaneous growth
The function 2ˣ is defined and differentiable for every real x. At x = 0 its value is 1 and its slope is ln(2) ≈ 0.6931, so its tangent line is y = 1 + x ln(2).
The ratio f′(x)/f(x) = ln(2) describes an instantaneous proportional rate. Over a full unit increase, the function doubles exactly: f(x + 1) = 2f(x). A derivative-based local approximation should not be confused with this finite change.
For a positive constant base a, the derivative is aˣ ln(a). Bases between zero and one give a negative derivative, base one gives zero, and bases greater than one give a positive derivative.
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.