Calculus Glossary

The rate of change of a function with respect to its variable. Geometrically, it represents the slope of the tangent line to the function's graph at a point.

The value that a function approaches as the input approaches some value. Fundamental to the definition of derivatives.

The process of finding the derivative of a function. Includes various rules and techniques for different function types.

A straight line that touches a curve at exactly one point. The derivative gives the slope of this line.

The steepness of a line. In calculus, the derivative gives the instantaneous slope of a curve at any point.

The rate of change at a specific point, given by the derivative at that point.

Rule for differentiating power functions: d/dx[xⁿ] = n·xⁿ⁻¹

Rule for differentiating composite functions: (f(g(x)))' = f'(g(x))·g'(x)

Rule for differentiating products: (f·g)' = f'·g + f·g'

Rule for differentiating quotients: (f/g)' = (f'·g - f·g')/g²

Technique for finding derivatives when functions are defined implicitly rather than explicitly.

Technique using logarithms to simplify differentiation of complex products/quotients.

Function of the form: aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀

Functions based on trigonometric ratios: sin(x), cos(x), tan(x), etc.

Function of the form: f(x) = a·bˣ or f(x) = eˣ

Inverse of exponential functions: ln(x), logₐ(x)

Ratio of two polynomial functions: f(x) = P(x)/Q(x)

Function of a function: f(g(x))

Rate of change of position with respect to time. First derivative of position function.

Rate of change of velocity with respect to time. Second derivative of position.

In economics, the derivative of the cost function. Approximate cost of producing one more unit.

Problems involving rates of change of related variables. Solved using implicit differentiation.

Finding maximum or minimum values using derivatives. Applications in business, physics, engineering.

Using derivatives to analyze and sketch function graphs: increasing/decreasing, concavity, extrema.

Derivative of a multivariable function with respect to one variable, holding others constant.

Rate of change of a multivariable function in a specific direction vector.

Vector of all partial derivatives of a multivariable function. Points in direction of greatest increase.

Infinite series representation of a function using its derivatives at a point.

Matrix of all first-order partial derivatives of a vector-valued function.

Matrix of second-order partial derivatives. Used in multivariable optimization.