Continuity, Differentiability and Applications

States the necessary and sufficient condition for a real-valued function to be continuous at an interior point.

Guarantees that the composition of two continuous functions is continuous at the specified point.

Ensures that a continuous function on a closed interval attains every intermediate value, guaranteeing a zero if endpoint signs differ.

States that a function is differentiable at a point if and only if both one-sided derivatives exist and are equal.

Establishes that differentiability at a point is a stronger condition than continuity and implies it directly.

Differentiates composite functions by taking the product of the outer and inner derivatives.

Gives standard derivatives of exponential and logarithmic functions for natural base e and general base a.

Standard derivative formulas for the principal inverse trigonometric functions.

Differentiates functions with variable bases and variable exponents by taking logarithms before differentiating.

Calculates the first derivative of y with respect to x when both variables are given parametrically in t.

Calculates the second derivative for parametrically defined curves by applying the chain rule to dy/dx.

Finds the derivative of an invertible function's inverse in terms of the original function's derivative.

Guarantees at least one stationary point for functions continuous on [a,b] and differentiable on (a,b) with equal endpoints.

Guarantees a point where the tangent is parallel to the secant line joining endpoints on a continuous and differentiable interval.

Determines the increasing or decreasing nature of a differentiable function on an interval from the sign of its derivative.

Gives the point-slope Cartesian equations of the tangent and normal lines to y = f(x) at (x1, y1).

Identifies local maxima and minima through the change of sign of f'(x) across a critical point.

Classifies stationary points as local maxima or local minima using the concavity given by the second derivative.