expKx

Expkx is not a widely standardized term on its own. In many mathematical and scientific contexts it represents the exponential function with the argument kx, written as exp(kx) or e^(kx), where k is a constant and x is the variable.

Definition and basic properties

Expkx equals e raised to the power of k times x: expkx = e^(k x). The derivative with respect to x is d/dx expkx = k e^(k x). The integral with respect to x is ∫ expkx dx = (1/k) e^(k x) + C for k ≠ 0; when k = 0, ∫ expkx dx = ∫ 1 dx = x + C. The value at x = 0 is expk0 = e^0 = 1 for any k.

Behaviour and interpretation

If k > 0, expkx grows exponentially as x increases; if k < 0, it decays. The function is monotone in x, with the rate of growth or decay controlled by the magnitude of k. Expkx often appears as a solution component in differential equations, for example as part of the general solution to dy/dx = k y, whose form is y = C e^(k x).

Context and usage

Expkx is commonly used to model processes with constant proportional growth or decay, including population models, chemical kinetics, and electrical RC-type responses when time is the independent variable. In numerical work, one should watch for floating-point overflow or underflow for large values of |k x|.

Note

The expression expkx is shorthand for exp(kx); exact meaning depends on context, but it typically denotes the exponential function of the product kx. See also the exponential function and e^(x).