Differential calculus

Differential calculus studies derivatives and how functions change locally. Its central tools describe instantaneous rates of change, slopes, and local approximations.

Derivative: A derivative measures a function’s instantaneous rate of change with respect to an input. The derivative is differential calculus’s central quantity.
Function (mathematics): A function assigns each input in its domain exactly one output. Differential calculus studies how function outputs respond to changing inputs.
Optimization (mathematics): Mathematical optimization seeks inputs that maximize or minimize an objective function under specified conditions. Derivatives locate candidate extrema and help test their behavior.
Integral calculus: Integral calculus studies accumulation and areas through integrals. It addresses accumulation rather than local rates, while remaining linked to derivatives by the fundamental theorem.
Limit (mathematics): A limit describes the value a function approaches as its input approaches a specified point. Limits define derivatives through increasingly small changes in input.
Slope: Slope measures the steepness and direction of a line, usually as vertical change divided by horizontal change. A derivative generalizes slope from straight lines to curves at a point.
Related rates: Related rates problems use derivatives to connect rates of change among quantities linked by an equation. They apply implicit differentiation to quantities changing together over time.
Finite difference: A finite difference measures change between function values at distinct inputs. Unlike a derivative, it uses a nonzero interval rather than an instantaneous limit.
Differential (mathematics): A differential is a linear approximation to the change in a function caused by a small change in its input. Differentials express derivative-based changes in a form useful for approximation.
Continuity (mathematics): A function is continuous at a point when its values approach its value there as the input approaches that point. Differentiability implies continuity, though continuity alone does not guarantee a derivative.

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