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Calculus › Limits & the derivative › derivative

derivative

Limits & the derivative

The derivative measures the instantaneous rate of change of the output variable with respect to the input variable. It provides a quantitative indication of how much the output of a function is expected to change for a small change in the input. The derivative serves as a tool for analyzing the sensitivity of one variable to changes in another.

Definition

The derivative of a function with respect to its input is defined as the limit of the ratio of the change in the function's value to the change in the input, as the change in the input approaches zero. This limit, if it exists, provides the instantaneous rate of change of the function at a given point.

f′(a)=limh→0f(a+h)-f(a)h

Geometric meaning

The geometric meaning of the derivative of a function at a fixed point is represented by the slope of the tangent line to the graph of the function at that point. This slope indicates the rate at which the function's value changes as the input changes, providing a visual interpretation of the derivative's role in calculus.

Rates and units

The derivative of a function represents the instantaneous rate of change of the output with respect to the input, and its units are determined by the units of the output divided by the units of the input. The units of the derivative are therefore expressed as units of the output per unit of the input. The concept of the derivative is inherently tied to the context and units of the original function.

A derivative from its definition

For the function f(x)=x-x2 , use the limit definition of the derivative to compute f′(2) .

f′(2)=limh→0f(2+h)-f(2)h
f(2)=2-22=-2
f(2+h)=(2+h)-(2+h)^2
f′(2)=limh→0(2+h)-(2+h)^2-(-2)h
f′(2)=limh→02+h-4-4h-h2+2h
f′(2)=limh→0-3h-h2h
f′(2)=limh→0(-3-h)
f′(2)=-3

Differentiability and continuity

Differentiability and continuity are closely related concepts in calculus, where differentiability implies continuity but not necessarily vice versa. A function is differentiable at a point if its derivative exists there, which requires the function to be continuous at that point and for the limit defining the derivative to exist. However, continuity alone is not sufficient for differentiability, as a function may be continuous at a point yet fail to have a derivative there due to the presence of a sharp corner, cusp, or vertical tangent.

Notes & references