differentiable function
[differentiable function|differentiable function] [differentiable function] is a function that has a derivative at each point in its domain.
Definition
differentiable function [differentiable function] is a function that has a derivative at each point in its domain. Rolle's Theorem states that if a differentiable function's outputs are equal at the endpoints of an interval, then there exists at least one point within the interval where the derivative is zero. This theorem applies to functions that are differentiable on an open interval and continuous on a closed interval.
Mechanism
differentiable function Theorem 3.15 establishes that for x > 0 and y = Inx, the derivative follows equation (3.30). More generally, when g(x) is a differentiable function, this principle extends. The tangent line equation for a differentiable function f at x = a provides a method to approximate f(x) near a. This mechanism relies on the derivative's role in capturing local behavior. The general applicability of this approach is supported by the equation's structure.
Causes
The Mean <a href='/en/entity/value-theorem'>Value Theorem</a> applies to differentiable function over an interval, ensuring the existence of a point c where the derivative equals the average rate of change. Since f(a) > f(b), the difference f(b) - f(a) is negative, indicating a decreasing function. This leads to the conclusion that such a c must satisfy the theorem's condition for the interval (a, b).
Effects
differentiable function A differentiable function's behavior influences the Mean <a href='/en/entity/value-theorem'>Value Theorem</a>'s applicability. When f(a) > f(b), the theorem ensures a point c exists in (a, b) where the derivative matches the slope between endpoints. This relationship reveals that a twice-differentiable function with negative second derivative is concave down.