Continuity and Differentiability

A function f(x) is continuous at

A function f(x) is continuous at point c if lim(x→c) f(x) = f(c). This means the left-hand limit, right-hand limit, and function value at c must all e

A function is differentiable at

A function is differentiable at a point if its derivative exists at that point. Every differentiable function is continuous, but not every continuous

For composite functions f = v

For composite functions f = v ∘ u, if t = u(x) and both dt/dx and dv/dt exist, then df/dx = (dv/dt) × (dt/dx). This rule is crucial for differentiatin

Standard derivatives include

Standard derivatives include: d/dx(sin⁻¹x) = 1/√(1-x²), d/dx(cos⁻¹x) = -1/√(1-x²), d/dx(tan⁻¹x) = 1/(1+x²). These functions have restricted domains an

Key derivatives

Key derivatives: d/dx(eˣ) = eˣ and d/dx(log x) = 1/x. The exponential function is its own derivative, making it unique and powerful in mathematical mo