Integrals and Differentials

Differential and integral calculus together describe how quantities change and accumulate. Differentiation finds the instantaneous rate of change of a function, geometrically the slope of the tangent to its graph. Standard rules make differentiation systematic: the power, product and quotient rules handle algebraic combinations, while the chain rule differentiates composite functions by multiplying the rates of change of the inner and outer functions. Implicit differentiation extends these methods to relations that are not written explicitly as one variable in terms of another.

Integration is the reverse operation. The indefinite integral, or antiderivative, recovers a function from its rate of change and is determined only up to an added constant. The definite integral measures accumulated quantity, such as the area under a curve, and the Fundamental Theorem of Calculus ties the two together. Multiple integrals extend integration to functions of several variables, computing volumes, masses and averages over regions of the plane or space; changing variables in such integrals introduces the Jacobian, a determinant that accounts for how areas or volumes are stretched.

Differential equations bring these ideas together by relating a function to its own derivatives. They model processes throughout science and engineering, from population growth and radioactive decay to the motion of mechanical systems and the flow of heat. Solving a differential equation means finding the functions that satisfy the relationship, and the techniques for doing so draw directly on the rules of differentiation and integration.

Frequently asked questions

What does the chain rule do?

It differentiates a composite function by multiplying the derivative of the outer function by the derivative of the inner function.

What is a multiple integral?

It is an integral over a function of several variables, used to find quantities such as areas, volumes and masses across two- or three-dimensional regions.

What is a differential equation?

It is an equation that relates a function to its derivatives, used to model how systems change; solving it means finding the functions that satisfy the relation.