Derivative of the Exponential Function

The difference quotient of eˣ can be rewritten as eˣ·(e^h − 1)/h, because e^(x+h) = eˣ·e^h. The factor (e^h − 1)/h does not depend on x; its limit as h → 0 is the slope of the exponential function at 0. The number e is defined precisely so that this limit equals 1: of all exponential functions aˣ, only the one with base e ≈ 2.71828 has slope 1 at 0. That is why differentiating simply leaves eˣ. For any other base, (aˣ)′ = ln(a)·aˣ, the factor ln a measuring how much the base deviates from e. This self-reproduction makes the exponential function the natural model for growth whose speed is proportional to the current amount.

Use the chain rule: the outer function is the exponential, whose derivative is itself, the inner function is kx with derivative k. So (e^(kx))′ = k·e^(kx); the factor from the exponent moves to the front. Concretely: (e^(2x))′ = 2e^(2x), (e^(−0.5x))′ = −0.5·e^(−0.5x). With a general expression in the exponent the same principle applies: (e^(u(x)))′ = u′(x)·e^(u(x)), for example (e^(x²))′ = 2x·e^(x²). The classic exam mistake is forgetting the inner derivative and writing only e^(2x); the result is then too small by the factor 2. Mnemonic: copy the exponential, multiply the inner derivative in front.

Rewrite the base: a^x = e^(x·ln a), since e^(ln a) = a. Now the chain rule applies with inner derivative ln a, so (a^x)′ = ln(a)·a^x. For 2^x this means (2^x)′ = ln(2)·2^x ≈ 0.693·2^x, for 10^x correspondingly ln(10)·10^x ≈ 2.303·10^x. The factor ln a is the growth rate: for a > e it exceeds 1, for a < e it is smaller. The formula also shows why e is the "natural" base: only for a = e does ln a = 1 and the prefactor disappear. Exams like to use 2^x as a trap, because many reflexively write 2^x or x·2^(x−1); both are wrong.

Here you combine product rule and chain rule. For f(x) = x·eˣ, f′(x) = 1·eˣ + x·eˣ = (1 + x)·eˣ. For g(x) = x²·e^(−x) the product rule gives g′(x) = 2x·e^(−x) + x²·(−1)·e^(−x) = (2x − x²)·e^(−x); the inner derivative −1 comes from the chain rule. Factoring out the exponential at the end is practical: since eˣ or e^(−x) is never zero, you read the zeros of the derivative from the polynomial factor alone, here 2x − x² = x(2 − x) with candidates x = 0 and x = 2. Exactly this pattern, polynomial times exponential, is the most common function type in final-exam curve analysis.

Because its derivative is proportional to itself: f(x) = c·e^(kx) satisfies f′ = k·f. Exactly this characterizes natural processes where the rate of change depends on the current amount: the more bacteria there are, the more are added per hour; the more radioactive nuclei exist, the more decay per second. Positive k means growth, negative k decay. The exponential family is the only class of functions with this property, which is why it solves the differential equation f′ = k·f. In applications you determine k from data such as doubling time (k = ln 2/T_D) or half-life (k = −ln 2/T_H). Thus the differentiation rule becomes a direct modelling tool for biology, physics and finance.