dcdx

dCdx, usually written as dC/dx, denotes the derivative of a quantity C with respect to the independent variable x. If C is a differentiable function of x, then dC/dx = lim_{h→0} [C(x+h) − C(x)]/h, representing the instantaneous rate of change of C with respect to x. It is the slope of the graph of C(x) at the point x and is often denoted C′(x).

In simple one-variable calculus, C is a scalar function of a single variable x. In multivariable contexts, the partial derivative ∂C/∂x measures the rate of change of C with respect to x while holding other variables fixed. If x affects other variables that also depend on x, the total derivative dC/dx is obtained via the chain rule and accounts for those dependencies.

Units of dC/dx are the units of C per unit of x. A positive dC/dx indicates C increases with x, while a negative value indicates a decrease. Common examples include C(x) = x^2, giving dC/dx = 2x, and C(x) = sin(x), giving dC/dx = cos(x).

Applications span mathematics and sciences. In physics and chemistry, dC/dx often represents a spatial gradient, as in diffusion where Fick’s first law relates flux to −D dC/dx. In numerical analysis, derivatives are approximated by finite difference formulas, such as the central difference [C(x+h) − C(x−h)]/(2h).

Limitations include the requirement that C be differentiable at the point of interest; where C is discontinuous or non-differentiable, dC/dx is undefined. See also gradient, derivative, and partial derivative.