251 Differential Calculus Terms and Definitions | Mathematics Board Exam Review

INTRODUCTION

Differential Calculus is the subject that separates reviewees who merely memorize formulas from those who actually understand what those formulas are doing. It is one of the most heavily tested topics on the PRC engineering licensure examination, and it shows up not just in the mathematics portion but embedded in problems across multiple engineering subjects. Rates of change, optimization, curve behavior, and approximation are trace back to the concepts you first encounter in Differential Calculus.

The vocabulary of Differential Calculus is unusually dense. A single problem can involve limits, continuity, differentiability, critical points, and concavity all at once. If any one of those terms is unclear to you, the entire solution falls apart. Many reviewees can execute the mechanical steps of differentiation: apply the power rule, use the chain rule, compute the derivative, but struggle when a problem asks for something conceptual, like identifying a point of inflection from a graph or explaining why a function fails to be differentiable at a given point.

This post covers 251 terms and definitions in Differential Calculus, alphabetically sorted and written with the PRC board exam in mind. Every definition is precise, board-relevant, and written in plain language. The goal is not to replace your textbook or your review manual. The goal is to give you a reliable, scannable reference that builds the conceptual vocabulary you need to read problems accurately and solve them confidently.

Work through this list systematically. Do not rush past terms that feel familiar. Some of the most commonly tested board exam items are built on terms that reviewees assume they know but have never fully examined. A solid command of these 251 definitions, combined with consistent practice on numerical problems, is one of the most efficient investments you can make in your review.

251 Differential Calculus Terms and Definitions

1. Abscissa

The horizontal coordinate of a point in the Cartesian plane, commonly denoted x. In curve tracing and function analysis, the abscissa identifies the input value at which a function is being evaluated.

2. Absolute Extremum

The highest or lowest value of a function over its entire domain. An absolute maximum is the largest output value the function ever attains, and an absolute minimum is the smallest. These are also called global extrema.

3. Absolute Maximum

The largest value of a function f(x) over its entire domain or over a specified closed interval. If f(c) is greater than or equal to f(x) for all x in the domain, then f(c) is the absolute maximum value.

4. Absolute Minimum

The smallest value of a function f(x) over its entire domain or over a specified closed interval. If f(c) is less than or equal to f(x) for all x in the domain, then f(c) is the absolute minimum value.

5. Absolute Value Function

The function defined as f(x) equals x when x is non-negative and equals negative x when x is negative. It is not differentiable at x equals zero because the left-hand and right-hand derivatives differ at that point.

6. Acceleration

The rate of change of velocity with respect to time. In calculus terms, acceleration is the second derivative of position with respect to time. It measures how quickly an object is speeding up or slowing down.

7. Acceleration Function

The second derivative of the position function with respect to time. It tells how quickly velocity is changing at each instant. Positive acceleration means the object is speeding up in the positive direction; negative acceleration means it is slowing down or moving faster in the negative direction.

8. Algebraic Function

A function that can be expressed using algebraic operations: addition, subtraction, multiplication, division, and rational exponents on the variable. Polynomial, rational, and radical functions are all algebraic functions.

9. Analytic Function

A function that is locally given by a convergent power series at every point in its domain. All elementary functions: polynomials, exponentials, trigonometric functions, and their inverses, are analytic on their domains.

10. Antecedent

In the context of a limit, the antecedent is the expression or condition that approaches a specific value. The term is occasionally used in formal logic statements involving limits and continuity.

11. Antiderivative

A function F(x) whose derivative equals a given function f(x). That is, F prime of x equals f(x). Finding antiderivatives is the central operation of integral calculus, but the concept arises in differential calculus when discussing the relationship between a function and its derivative.

12. Antidifferentiation

The reverse process of differentiation. Given f prime of x, antidifferentiation produces f(x). It is the gateway concept linking differential calculus to integral calculus. All antiderivatives of a function differ only by a constant.

13. Approximation by Differentials

A technique using the differential of a function to estimate small changes in the function’s value. If y equals f(x), then the change in y is approximately dy equals f prime of x times dx, where dx is a small change in x.

14. Arc Length

The length of a curve between two points. In differential calculus, the arc length element ds is defined using the first derivative of the curve’s equation. The full arc length is computed by integrating the arc length element.

15. Arc Length Differential

The infinitesimal element of arc length along a curve, denoted ds. It equals the square root of the sum of dx squared and dy squared. For a function y equals f(x), it can be written as the square root of one plus the square of f prime of x, times dx.

16. Asymptote

A line that a curve approaches but never reaches as the variable tends to infinity or to some finite value. There are three types: vertical, horizontal, and oblique. Asymptotes are key features identified during curve tracing.

17. Asymptotic Behavior

The behavior of a function as the input grows without bound or approaches a specific finite value. Asymptotic behavior is described using limits and is used to identify horizontal and vertical asymptotes during curve analysis.

18. Average Rate of Change

The change in the value of a function divided by the change in the input over a specified interval. For f(x) over the interval from a to b, the average rate of change is f(b) minus f(a), divided by b minus a. It equals the slope of the secant line through the two points.

19. Average Value of a Function

The mean output value of a function over a closed interval. It is computed as one over the length of the interval, times the integral of the function over that interval. This concept bridges differential and integral calculus.

20. Average Velocity

The total displacement divided by the total time elapsed. It is the slope of the secant line on a position-time graph. As the time interval shrinks to zero, the average velocity approaches the instantaneous velocity.

21. Bifurcation Point

A point at which a small change in a parameter causes a sudden qualitative change in the behavior of a function or system. In calculus, it is associated with changes in the number or nature of critical points.

22. Boundary Value

A value of the independent variable at the endpoint of a closed interval. In optimization problems on closed intervals, boundary values must be checked along with critical points to find absolute extrema.

23. Bounded Function

A function whose output values are confined between some fixed lower and upper bounds for all inputs in the domain. A continuous function on a closed interval is always bounded, by the Extreme Value Theorem.

24. Bounded Variation

A property of a function whose total variation, the sum of absolute changes in function value, is finite over a given interval. Functions of bounded variation are differentiable almost everywhere.

25. Calculus

The branch of mathematics concerned with rates of change and accumulation. It is divided into differential calculus, which deals with derivatives and rates of change, and integral calculus, which deals with integrals and accumulation of quantities.

26. Catenary

The curve formed by a flexible chain hanging freely between two supports under its own weight. Its equation is y equals a times the hyperbolic cosine of x over a. It is a classic example in differential calculus involving hyperbolic functions.

27. Cauchy’s Mean Value Theorem

A generalization of the Mean Value Theorem stating that for two functions f and g that are continuous on a closed interval and differentiable on the open interval, there exists a point c such that f prime of c times the change in g equals g prime of c times the change in f.

28. Chain Rule

A differentiation rule for composite functions. If y equals f(g(x)), then dy over dx equals f prime of g(x) times g prime of x. It is one of the most frequently used differentiation rules in both pure and applied calculus problems.

29. Change of Variable

A substitution technique in which a new variable is introduced to simplify a function or expression before differentiation. It is the formal basis for the chain rule and for substitution in integration.

30. Characteristic Equation

In the context of differential equations derived from calculus, the characteristic equation is formed by substituting an exponential trial solution into a linear differential equation. Its roots determine the form of the general solution.

31. Closed Interval

An interval that includes both of its endpoints, written as the set of all x such that a is less than or equal to x and x is less than or equal to b. Written in bracket notation as [a, b]. Continuity on a closed interval enables the Extreme Value Theorem and the Mean Value Theorem.

32. Cofunction

A trigonometric function that is complementary to another. Sine and cosine are cofunctions, as are tangent and cotangent, and secant and cosecant. The derivative of a cofunction follows a predictable sign pattern relative to the derivative of its paired function.

33. Composite Function

A function formed by applying one function to the output of another. If f and g are functions, the composite f of g of x applies g first and then f. Differentiating composite functions requires the chain rule.

34. Concave Down

A description of a curve that opens downward over an interval, resembling an inverted bowl. A function is concave down where its second derivative is negative. A local maximum often occurs at a point where the curve transitions from concave up to concave down.

35. Concave Up

A description of a curve that opens upward over an interval, resembling a right-side-up bowl. A function is concave up where its second derivative is positive. A local minimum often occurs at a point where the curve transitions from concave down to concave up.

36. Concavity

The property of a curve that describes whether it bends upward or downward. Concavity is determined by the sign of the second derivative. Knowing the concavity of a curve at a critical point helps classify that point as a local maximum or minimum.

37. Concavity Test

The use of the second derivative to determine whether a function is concave up or concave down on an interval. If the second derivative is positive on an interval, the function is concave up. If negative, it is concave down.

38. Condition of Continuity

The three conditions that must all be satisfied for a function f to be continuous at a point x equals c: the function value f(c) must exist, the limit of f(x) as x approaches c must exist, and that limit must equal f(c).

39. Constant Function

A function whose output is the same for all inputs. Its derivative is zero everywhere. On a graph, it appears as a horizontal line.

40. Constant Multiple Rule

A differentiation rule stating that the derivative of a constant times a function equals the constant times the derivative of the function. That is, the derivative of k times f(x) is k times f prime of x, where k is a constant.

41. Constant of Differentiation

A concept that arises in implicit differentiation and related rates, referring to any constant term in an equation whose derivative is zero. It is distinct from the constant of integration, which arises in antidifferentiation.

42. Continuity

A property of a function at a point, meaning the function has no breaks, holes, or jumps at that point. A function f is continuous at x equals c if the limit of f(x) as x approaches c equals f(c), and both exist. Continuity is a prerequisite for differentiability.

43. Continuity on a Closed Interval

A function is continuous on a closed interval [a, b] if it is continuous at every interior point and the one-sided limits at the endpoints match the function values. This condition is required by the Extreme Value Theorem and the Mean Value Theorem.

44. Continuous Extension

The process of redefining a function at a removable discontinuity to make it continuous at that point. The redefined value is set equal to the limit at the point of discontinuity.

45. Continuous Function

A function that is continuous at every point in its domain. Its graph can be drawn without lifting the pen. Polynomial functions, exponential functions, and trigonometric functions are all continuous on their natural domains.

46. Continuous on an Open Interval

A function that is continuous at every point in an open interval (a, b). It does not require the function to be defined or continuous at the endpoints. Most differentiable functions are continuous on open intervals.

47. Contrapositive

In calculus logic, the contrapositive of “if f is differentiable at c, then f is continuous at c” is “if f is not continuous at c, then f is not differentiable at c.” The contrapositive is always logically equivalent to the original statement.

48. Corner Point

A point on a graph where the curve has a sharp turn and the left-hand and right-hand derivatives exist but are not equal. A function is not differentiable at a corner point. An example is the absolute value function at x equals zero.

49. Critical Number

A value of x in the domain of f where either f prime of x equals zero or f prime of x does not exist. Critical numbers are the candidates for local extrema and are the first step in optimization using calculus.

50. Critical Point

A point on the graph of a function corresponding to a critical number. At a critical point, the tangent line is either horizontal or undefined. Critical points must be tested to determine whether they are local maxima, local minima, or neither.

51. Curvature

A measure of how sharply a curve bends at a given point. It is defined as the rate of change of the tangent direction with respect to arc length. For a function y equals f(x), the curvature formula involves both the first and second derivatives.

52. Curve Sketching

The process of using calculus tools: intercepts, asymptotes, critical points, concavity, and inflection points to draw an accurate graph of a function. It is a systematic technique that appears frequently in board exam problems.

53. Curve Tracing

The process of plotting a curve point by point using specific values or parametric equations, then identifying key features such as symmetry, loops, cusps, and asymptotes. It is closely related to curve sketching but often refers to more complex or parametric curves.

54. Cusp

A point on a curve where the tangent line is vertical and the curve reverses direction abruptly. At a cusp, the function is typically continuous but not differentiable. Cusps are identified by the behavior of the derivative near the point.

55. Decreasing Function

A function whose output values decrease as the input values increase over some interval. Formally, f is decreasing on an interval if f prime of x is negative for all x in that interval.

56. Decreasing on an Interval

A function is decreasing on an interval if its first derivative is negative throughout that interval. The function loses value as x increases through the interval.

57. Definite Integral (as a Limit)

The limit of a Riemann sum as the number of subintervals approaches infinity and the width of each subinterval approaches zero. While primarily an integral calculus concept, the definite integral as a limit connects back to the foundational ideas of differential calculus.

58. Delta-Epsilon Definition of a Limit

The formal mathematical definition of a limit. It states that the limit of f(x) as x approaches c equals L if, for every positive number epsilon, there exists a positive number delta such that whenever x is within delta of c (but not equal to c), f(x) is within epsilon of L.

59. Dependent Rate

In a related rates problem, the rate of change that is being solved for. It is found by differentiating the equation relating the quantities with respect to time and substituting the known rates and values.

60. Dependent Variable

The variable whose value is determined by the input. In y equals f(x), y is the dependent variable. Its rate of change with respect to the independent variable is what the derivative measures.

61. Derivative

The instantaneous rate of change of a function with respect to its independent variable. It is defined as the limit of the difference quotient as the change in x approaches zero. Geometrically, it equals the slope of the tangent line to the curve at a given point.

62. Derivative of a Composite Function

The derivative of f(g(x)), computed using the chain rule. It equals f prime evaluated at g(x), multiplied by g prime of x. This is one of the most important and frequently tested differentiation rules.

63. Derivative of a Constant

The derivative of any constant function is zero. This is because a constant function does not change, so its rate of change is zero everywhere.

64. Derivative of a Product

The derivative of the product of two functions, computed using the product rule. It equals the first function times the derivative of the second, plus the second function times the derivative of the first.

65. Derivative of a Quotient

The derivative of the quotient of two functions, computed using the quotient rule. It equals the denominator times the derivative of the numerator, minus the numerator times the derivative of the denominator, all divided by the square of the denominator.

66. Derivative of an Exponential Function

The derivative of e to the x is e to the x. The derivative of a to the x, where a is a positive constant, is a to the x times the natural logarithm of a.

67. Derivative of an Inverse Function

If f and g are inverse functions and f prime of g(x) is not zero, then g prime of x equals one divided by f prime of g(x). This relationship is used to differentiate inverse trigonometric and inverse exponential functions.

68. Derivative of a Logarithmic Function

The derivative of the natural logarithm of x is one over x. The derivative of the logarithm base a of x is one divided by the product of x and the natural logarithm of a.

69. Derivative of a Trigonometric Function

The standard derivatives of the six trigonometric functions: the derivative of sine x is cosine x; of cosine x is negative sine x; of tangent x is secant squared x; of cotangent x is negative cosecant squared x; of secant x is secant x times tangent x; and of cosecant x is negative cosecant x times cotangent x.

70. Derivative Test

A method using the sign or value of the first or second derivative to classify critical points as local maxima, local minima, or saddle points. The First Derivative Test and the Second Derivative Test are the two standard forms.

71. Derived Function

An older term for the derivative of a function. The derived function of f is f prime, giving the instantaneous rate of change at every point in the domain.

72. Difference Quotient

The expression f(x plus h) minus f(x), divided by h. It represents the average rate of change of f over the interval from x to x plus h. The derivative is the limit of the difference quotient as h approaches zero.

73. Differentiability

The property of a function at a point where the derivative exists. A function is differentiable at x equals c if the limit of the difference quotient exists at that point. Differentiability implies continuity, but continuity does not imply differentiability.

74. Differentiable Function

A function that has a derivative at every point in its domain, or over a specified interval. Polynomial functions, exponential functions, and sine and cosine are differentiable everywhere. Functions with corners, cusps, or vertical tangents are not differentiable at those points.

75. Differential

An infinitesimally small change in a variable. The differential of y, written dy, is defined as f prime of x times dx, where dx is an infinitesimally small change in x. Differentials are used in linear approximations and in the formalism of integration.

76. Differential Calculus

The branch of calculus concerned with the study of derivatives, rates of change, and the behavior of functions. It includes the theory of limits, the definition and computation of derivatives, and applications such as optimization and curve sketching.

77. Differential Coefficient

An older term for the derivative of a function, used in classical calculus texts. The differential coefficient of y with respect to x is written dy over dx and is equivalent to f prime of x.

78. Differential Equation

An equation that involves a function and one or more of its derivatives. Differential equations describe many physical phenomena and are central to engineering analysis. Their study begins with the concepts established in differential calculus.

79. Differentials, Exact

A differential expression M dx plus N dy is called exact if it is the total differential of some function f(x, y), meaning the partial derivative of f with respect to x equals M and the partial derivative with respect to y equals N.

80. Differentiation

The process of computing the derivative of a function. It involves applying differentiation rules: power rule, product rule, quotient rule, chain rule, and others  to find the instantaneous rate of change of a function.

81. Differentiation Formulas

The standard rules and results used to compute derivatives. These include the power rule, the derivative of a constant, the sum rule, the product rule, the quotient rule, the chain rule, and the derivatives of standard functions like exponentials, logarithms, and trigonometric functions.

82. Direction of Curve

The direction in which a curve is traced as the parameter or independent variable increases. The direction of the curve at any point is along the tangent vector at that point, in the direction of increasing parameter.

83. Discontinuity

A point at which a function fails to be continuous. Types include removable discontinuities, jump discontinuities, and infinite discontinuities. Identifying discontinuities is an important step in limit analysis and function behavior.

84. Discontinuity of the First Kind

Another name for a jump discontinuity. At a discontinuity of the first kind, both one-sided limits exist as finite numbers but are unequal. The function has a definite jump at that point.

85. Discontinuity of the Second Kind

A discontinuity at which at least one of the one-sided limits either does not exist or is infinite. Infinite discontinuities and oscillating discontinuities are examples. They are more severe than removable or jump discontinuities.

86. Discontinuous Function

A function that is not continuous at one or more points in its domain. Examples include piecewise functions with gaps, rational functions with holes or vertical asymptotes, and functions with jump behavior.

87. Displacement

The change in position of a moving object. It is the integral of velocity with respect to time, but in differential calculus, it is the quantity whose derivative gives velocity. Displacement is a signed quantity, unlike distance.

88. Domain

The set of all input values for which a function is defined. Determining the domain is the first step in analyzing any function. Restrictions on the domain arise from denominators, square roots, and logarithms.

89. Double Root

A root of a function where the factor appears twice in the factored form. At a double root, the graph of the function touches the x-axis but does not cross it. The derivative at a double root is zero.

90. Dummy Variable

A variable used in a mathematical expression that can be replaced by any other variable without changing the meaning of the expression. In the limit definition of a derivative, the variable that approaches zero is a dummy variable.

91. e (Euler’s Number)

The base of the natural logarithm, approximately equal to 2.71828. It is defined as the limit of the quantity one plus one over n, raised to the power n, as n approaches infinity. The function e to the x is the unique function equal to its own derivative.

92. Elasticity

In applied calculus, the elasticity of a function measures the percentage change in output relative to a percentage change in input. It is expressed in terms of the derivative and the function values.

93. Elasticity of Demand

An application of differential calculus in economics. It measures how responsive the quantity demanded is to a change in price and is defined using the derivative of the demand function. An elasticity greater than one in absolute value means the demand is elastic.

94. Element of Arc

The infinitesimal element of arc length along a curve. For a curve y equals f(x), the element of arc ds equals the square root of one plus the square of dy over dx, times dx.

95. Endpoint Extremum

A maximum or minimum value of a function that occurs at an endpoint of a closed interval. Endpoint extrema must be considered when finding absolute extrema on closed intervals and are not detected by setting the derivative equal to zero.

96. Envelope of a Family of Curves

A curve that is tangent to every member of a given family of curves. It is found by eliminating the parameter from the family’s equation and the equation obtained by differentiating with respect to the parameter.

97. Epsilon-Delta

The formal language of limits in calculus. An epsilon-delta proof rigorously establishes that the limit of f(x) as x approaches c equals L by showing that for every tolerance epsilon, a suitable proximity delta can be found. It is the foundation of rigorous calculus.

98. Equation of the Normal

The equation of the line perpendicular to the tangent to a curve at a given point. Its slope is the negative reciprocal of the derivative at that point, and it passes through the point of tangency.

99. Equation of the Tangent

The equation of the tangent line to a curve at a given point. Using the point-slope form with the slope equal to the derivative at the point, it is one of the most direct applications of the derivative concept.

100. Error in Approximation

The difference between an exact value and an approximate value obtained by a method such as linear approximation or differentials. In differential calculus, the error in linear approximation is related to the second derivative and the size of the increment.

251. Zero of a Function

A value of x for which f(x) equals zero. Zeros are the x-coordinates of the x-intercepts of the graph. Newton’s Method is a calculus-based technique for numerically approximating zeros.

CONCLUSION

Differential Calculus rewards those who understand it structurally, not just mechanically. The terms in this list span the full landscape of the subject from the foundational ideas of limits and continuity, through the mechanics of differentiation rules, and into the rich territory of applications like optimization, related rates, curve tracing, and motion analysis. When you know this vocabulary deeply, you can read a board exam problem and immediately recognize what tools to use and why. That recognition is what separates a confident solver from a hesitant one.

For the Math board exam, the areas that consistently produce the most problems are derivatives of standard functions, implicit differentiation, related rates, and optimization. Make sure you can apply the chain rule, the product rule, and the quotient rule without slowing down. Know the derivatives of all six trigonometric functions, the exponential function, and the natural logarithm cold. Be comfortable with implicit differentiation because it appears not just in its own problem type but embedded in related rates and curve analysis.

Give special attention to L’Hopital’s Rule, the Mean Value Theorem, the First and Second Derivative Tests, and the behavior of functions at critical points and inflection points. These are the conceptual pillars of the subject, and board exam items frequently test them in combination, asking you not just to compute but to reason about what the computation means. Review this list regularly, work through a large volume of practice problems, and always connect the definition of a term to its geometric or physical interpretation. That connection is what makes calculus truly useful in engineering practice.

For practice problems on all these topics, head over to our Differential Calculus Problems and Solutions section here on PinoyBix. Hundreds of solved exam-type questions, complete with step-by-step solutions, organized by topic so you can drill exactly what you need to work on.