1 Definition

The derivative describes how a quantity changes in response to changes in its input. In calculus, it is used to measure the local behavior of a function near a point, especially its rate of change and slope. The concept is central to differential calculus and appears throughout mathematics, physics, engineering, and economics.

1.1 Intuitive meaning

Intuitively, a derivative tells how fast a function is changing at a particular point. If a graph rises steeply, the derivative is large and positive; if it falls, the derivative is negative; and if it is flat, the derivative is near zero. This local measure is often interpreted as the best linear description of the function at that point.

1.2 Limit definition

The derivative at a point is defined using a limit that compares the change in the function value with the change in the input. This precise formulation captures an instantaneous rate rather than an average over an interval.

1.2.1 Difference quotient

For a function f, the difference quotient is the ratio

f(x + h) - f(x) / h

for a small nonzero change h. It represents the average rate of change over the interval from x to x + h. Taking the limit as h approaches 0 gives the derivative, provided the limit exists.

1.2.2 Instantaneous rate of change

The derivative is often described as the instantaneous rate of change. In physical terms, this means the rate at a single moment rather than over a span of time. In geometric terms, it gives the slope of the graph at one point.

1.3 Notation

Several notations are used for derivatives, each emphasizing a different aspect of the concept. These conventions are common in algebra, analysis, and applied mathematics.

1.3.1 Leibniz notation

Leibniz notation writes the derivative of y with respect to x as dy/dx. It is especially useful when the relationship between variables matters, and it is widely used in calculus and differential equations.

1.3.2 Prime notation

Prime notation writes the derivative of f as f'. This form is concise and common when working with single-variable functions, especially in elementary calculus.

1.3.3 D notation

D notation uses an operator such as Df or d/dx. It emphasizes differentiation as a process applied to a function and is often convenient in formal calculations.

2 Geometric interpretation

The derivative has a natural geometric meaning as the slope of a curve at a point. It connects algebraic formulas with the shape of graphs and helps describe local behavior visually.

2.1 Slope of a tangent line

At a smooth point on a graph, the derivative equals the slope of the tangent line. This line touches the curve and matches its direction at that point. The tangent slope summarizes how steep the graph is locally.

2.2 Secant lines and limiting process

A secant line passes through two points on a curve and has a slope based on the average change between them. As the second point moves closer to the first, the secant line approaches the tangent line. The derivative arises from this limiting process.

2.3 Local linear approximation

Near a differentiable point, a function can be approximated by a line. This linear approximation uses the function value and derivative at that point to estimate nearby values. It is one of the most useful practical consequences of differentiability.

3 Differentiability

Differentiability describes whether a derivative exists at a point or on an interval. A function may be continuous without being differentiable, but differentiability implies a strong degree of regularity.

3.1 Differentiable functions

A function is differentiable at a point if its derivative exists there. Such functions behave smoothly enough to admit a well-defined tangent line and local linear model.

3.2 Continuity and differentiability

Differentiability implies continuity, but continuity alone does not guarantee differentiability. A function can have no breaks or jumps and still fail to have a derivative at certain points because of sharp changes in direction or other irregular features.

3.3 Non-differentiable points

Some points do not admit a derivative. These are often associated with abrupt changes in shape or an infinite slope.

3.3.1 Corners

A corner occurs when the graph changes direction sharply, as in the absolute value function at the origin. The left-hand and right-hand slopes do not agree, so the derivative does not exist.

3.3.2 Cusps

A cusp is a pointed singularity where the graph comes to a sharp tip. The slope may change without settling to a finite value, preventing differentiability.

3.3.3 Vertical tangents

A vertical tangent has an undefined finite slope because the tangent line is vertical. In such cases, the derivative does not exist as a real number, even though the graph may still appear smooth.

4 Rules of differentiation

Differentiation rules make it possible to compute derivatives efficiently without returning to the limit definition each time. These rules apply to many standard expressions and are fundamental tools in calculus.

4.1 Constant rule

The derivative of a constant is zero. A constant function does not change as its input changes, so its rate of change is nil.

4.2 Power rule

The power rule states that the derivative of x^n is n x^(n-1) for suitable exponents. It is one of the most frequently used formulas in elementary calculus.

4.3 Sum and difference rules

The derivative of a sum is the sum of the derivatives, and the derivative of a difference is the difference of the derivatives. These rules reflect the linearity of differentiation.

4.4 Product rule

The product rule gives the derivative of a product of two functions. It combines the derivatives of each factor in a specific way and is essential when differentiating expressions formed by multiplication.

4.5 Quotient rule

The quotient rule is used for ratios of differentiable functions. It expresses the derivative of a quotient in terms of the derivatives of the numerator and denominator.

4.6 Chain rule

The chain rule differentiates composite functions. It states that the derivative of a composition is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.

4.7 Higher-order derivatives

A function may be differentiated more than once. The second derivative measures how the first derivative changes, and further derivatives continue the process. Higher-order derivatives are used in curvature analysis, approximation, and differential equations.

5 Derivatives of common functions

Standard function families have well-known derivatives. These formulas are widely used as building blocks in computation and theory.

5.1 Polynomial functions

Polynomial functions are differentiable everywhere on their domains, and their derivatives are also polynomials. Each term is handled by the power rule, making these derivatives straightforward to compute.

5.2 Rational functions

Rational functions are quotients of polynomials. Their derivatives are obtained using the quotient rule or by rewriting the expression when possible. They may fail to be differentiable where the denominator is zero.

5.3 Exponential functions

Exponential functions have derivatives proportional to themselves. This self-replicating behavior makes them especially important in models of growth and decay.

5.4 Logarithmic functions

Logarithmic functions differentiate to reciprocal-type expressions. They are closely related to exponential functions and often appear in problems involving scaling and multiplicative change.

5.5 Trigonometric functions

Trigonometric functions such as sine and cosine have derivatives that cycle among familiar forms. They are central in geometry, wave behavior, and periodic motion.

5.6 Inverse trigonometric functions

Inverse trigonometric functions also have standard derivative formulas. These expressions are useful when angles are recovered from ratios or coordinates.

6 Applications

Derivatives are used to analyze change in many settings. They provide tools for describing motion, finding extreme values, studying graph behavior, and approximating functions.

6.1 Motion in one dimension

In one-dimensional motion, derivatives connect position, velocity, and acceleration. They translate a position function into quantities that describe movement over time.

6.1.1 Velocity

Velocity is the derivative of position with respect to time. It gives the instantaneous speed and direction of motion along a line.

6.1.2 Acceleration

Acceleration is the derivative of velocity, or the second derivative of position. It measures how quickly velocity changes over time.

6.2 Optimization

Optimization uses derivatives to identify values that make a function as large or as small as possible. This is a major application in science, engineering, and economics.

6.2.1 Critical points

Critical points occur where the derivative is zero or undefined. They are candidates for local maxima, local minima, or other special behavior.

6.2.2 Maximum and minimum values

By analyzing derivatives near critical points and at boundaries, one can determine where a function reaches its highest or lowest values on a given domain.

6.3 Curve sketching

Derivatives help reveal the shape of a graph without plotting every point. They indicate where a function rises or falls and how its curvature changes.

6.3.1 Increasing and decreasing intervals

A function is increasing where its derivative is positive and decreasing where its derivative is negative. These intervals show the direction of change across a graph.

6.3.2 Concavity

Concavity describes whether a graph bends upward or downward. The second derivative often provides the main test for this behavior.

6.3.3 Inflection points

An inflection point is where concavity changes sign. These points mark a transition in the graph’s bending behavior.

Related rates problems involve several quantities connected by an equation, each changing with respect to time. Differentiation is used to relate their rates and solve for an unknown speed of change.

6.5 Linearization and differentials

Linearization uses the derivative to approximate a function near a point. Differentials provide a related way to estimate small changes in output from small changes in input.

7 Advanced topics

Beyond basic single-variable calculus, derivatives extend to more complex settings. These extensions are essential in higher mathematics and applied analysis.

7.1 Implicit differentiation

Implicit differentiation is used when a relation is given without solving explicitly for one variable. By differentiating both sides with respect to the chosen variable, one can find derivatives of quantities defined indirectly.

7.2 Parametric differentiation

Parametric differentiation treats variables defined by a parameter, often time. It allows the derivative of one coordinate with respect to another to be computed from their separate parameterized forms.

7.3 Partial derivatives

Partial derivatives measure change with respect to one variable while holding others fixed. They are fundamental in multivariable calculus and appear in models with several inputs.

7.4 Derivative as a linear map

In higher mathematics, the derivative at a point can be viewed as a linear map that best approximates a function near that point. This perspective generalizes the slope of a line to more abstract settings.

7.5 Higher-dimensional generalizations

Derivatives extend to functions of several variables and to mappings between vector spaces. These generalizations include gradients, Jacobians, and other tools that describe local change in multiple directions.

Several theorems explain why derivatives behave as they do and connect them to broader ideas in calculus. These results give structure to the theory and support many applications.

8.1 Mean value theorem

The mean value theorem states that under suitable conditions, a differentiable function has at least one point where its derivative equals its average rate of change over an interval. It links local and global behavior.

8.2 Rolle's theorem

Rolle's theorem is a special case of the mean value theorem. If a function has equal values at two endpoints and is differentiable in between, then its derivative is zero at some interior point.

8.3 Taylor's theorem

Taylor's theorem expresses a function as a polynomial approximation plus an error term. Derivatives at a point determine the coefficients of the approximating polynomial, making the theorem a powerful tool for estimation.

8.4 Relationship to integration

Differentiation and integration are closely connected through the fundamental theorem of calculus. In many settings, integration can recover accumulated change from a derivative, and differentiation can retrieve the rate from a cumulative quantity.

9 Historical development

The concept of derivative emerged from earlier ideas about motion, tangency, and change. Its formal development was a major step in the creation of calculus.

9.1 Early ideas of change

Before modern calculus, mathematicians studied slopes, motion, and infinitesimal quantities in geometric and physical contexts. These efforts laid the groundwork for a systematic theory of change.

9.2 Newton and Leibniz

Isaac Newton and Gottfried Wilhelm Leibniz developed the core ideas of calculus in the late 17th century. Newton emphasized motion and fluxions, while Leibniz introduced notation that strongly influenced modern calculus.

9.3 Modern formulation

Later mathematicians provided rigorous limit-based foundations for derivatives. This modern formulation clarified when derivatives exist and made calculus compatible with precise analysis.