The relationship and the definition of derivatives and anti-derivatives is described in this video. First, consider a function F(x)=x^3-5x+2 and another with small 'f', f(x)=3x-5. f(x) here is the derivative of F(x). However, on the contrary, F(x) is said to be the anti-derivative of f(x). There is a catch though; even though F(x) has only one derivative in the form of f(x), f(x) here has more than one anti-derivative. This is because f(x)'s anti-derivatives are of the form F(x)=x^3-5x+c, where c stands for any constant number. For example, f(x) anti-derivatives can be any of the following: F(x)=x^3-5x+2, F(x)=x^3-5x, F(x)=x^3-5x+4... and so on.
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