tuletise

Tuletise, or derivative, is a fundamental concept in calculus describing the instantaneous rate of change of a function with respect to its input. For a real-valued function f defined on an interval, the derivative at a point x is the limit f'(x) = lim_{h->0} (f(x+h) - f(x))/h, if it exists. Geometrically, it is the slope of the tangent line to the graph of f at x.

Notationally, derivatives are written f'(x) or df/dx; for multivariable functions, partial derivatives ∂f/∂x generalize to higher dimensions, forming gradients and Jacobians.

Existence: If f is differentiable at x, it is continuous there; however, a function can be continuous but not differentiable at a point (e.g., cusp). Differentiability implies local linear approximation by the tangent.

Rules: The derivative obeys several rules: the power rule (d/dx x^n = n x^{n-1}), the sum rule, product rule, quotient rule, and chain rule (d/dx f(g(x)) = f'(g(x)) g'(x)). Higher-order derivatives exist by differentiating again, f''(x), interpreted as acceleration in physics.

Examples: d/dx x^2 = 2x; d/dx sin x = cos x; d/dx e^x = e^x.

Applications: Derivatives measure rate of change, determine tangents, optimize functions, model motion, and analyze curves. In multiple variables, gradients point in the direction of steepest ascent.

Extensions: Numerical differentiation approximates derivatives; in higher dimensions, the gradient, Hessian, and Jacobian are used.

This is a concise overview of tuletise, highlighting its definition, notation, rules, and applications.

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