Difference Between Derivative and Integral

Derivative vs Integral

Differentiation and integration are two fundamental operations in Calculus. They have numerous applications in several fields, such as Mathematics, engineering and Physics. Both derivative and integral discuss the behavior of a function or behavior of a physical entity that we are interested about.

What is Derivative?

Suppose y=ƒ(x) and x₀ is in the domain of ƒ. Then lim_Δx→∞Δy/Δx = lim_Δx→∞[ƒ(x₀+Δx) − ƒ(x₀)]/Δx is called the instantaneous rate of change of ƒ at x₀, provide this limit exists finitely. This limit is also called the derivative of at  and is denoted by ƒ(x).

The value of the derivative of a function f at an arbitrary point x in the domain of the function is given by lim_Δx→∞[ƒ(x+Δx) − ƒ(x)]/Δx. This is denoted by any one of the following expressions: y, ƒ(x), ƒ, dƒ(x)/dx, dƒ/dx, D_xy.

For functions with several variables, we define partial derivative. The partial derivative of a function with several variables is its derivative with respect to one of those variables, assuming that the other variables are constants. The symbol of the partial derivative is ∂.

Geometrically the derivative of a function can be interpreted as the slope of the curve of the function ƒ(x).

What is Integral?

Integration or anti-differentiation is the reverse process of differentiation. In other words, it is the process of finding an original function when the derivative of the function is given. Therefore, an integral or an anti-derivative of a function ƒ(x) if, ƒ(x)=F(x) can be defined as the function F(x), for all x in the domain of ƒ(x).

The expression ∫ƒ(x) dx denotes the derivative of function ƒ(x). If ƒ(x)=F(x), then ∫ƒ(x) dx = F(x)+C, where C is a constant, ∫ƒ(x) dx is called the indefinite integral of ƒ(x).

For any function ƒ, which is not necessarily non-negative, and defined on the interval [a,b], _a∫^bƒ(x) dx is called the definite integral ƒ on [a,b].

The definite integral _a∫^bƒ(x) dx of a function ƒ(x) can be geometrically interpreted as the area of the region bounded by the curve ƒ(x) , the x-axis, and the lines x=a and x=b.