What is an Antiderivative?

An antiderivative, also known as an indefinite integral, is a function that has the property that its derivative is equal to the original function. In other words, if we take the derivative of an antiderivative, we get back the original function.

For example, consider the function f(x) = x^2. The antiderivative of f(x) is F(x) = x^3/3. To see why this is true, we can use the definition of a derivative:

F'(x) = d/dx [F(x)]

Using the chain rule, we have:

F'(x) = d/dx [x^3/3]

= d/dx (x^2)

= 2x

So, F'(x) = 2x, which is the same as the derivative of f(x). Therefore, F(x) is an antiderivative of f(x).

Antiderivatives are important in calculus because they allow us to integrate functions and find the area under curves. They also have many practical applications in fields such as physics, engineering, and economics.