The constant multiple rule states that for a constant C and a differentiable function f(x), the derivative of C*f(x) equals C times the derivative of f(x), allowing constant factors to be pulled outside a differentiation operation. This belongs to differential calculus, within the foundational rules of differentiation derived directly from the limit definition of the derivative, and it can be proven algebraically (by factoring the constant out of the difference-quotient limit) or understood geometrically (a constant multiple vertically scales a function's graph, scaling the slope of every secant and tangent line by the same factor).
The constant multiple rule states that for a constant C and a differentiable function f(x), the derivative of C*f(x) equals C times the derivative of f(x), allowing constant factors to be pulled outs…