In school, we’re taught about derivatives of natural number order. For example, we have that the first derivative of e²ˣ is just 2e²ˣ. If we take the derivative again, we find that we get 4e²ˣ. In any case, we are still taking the derivative an integral amount of times. So what does it mean to take the derivative say, 0.5 times, or π times? In this article, we explore exactly that.
Let us first look at the function x², for simplicity. When we differentiate this, we get 2x; we’re just bringing the exponent down and subtracting one. If we were to differentiate xⁿ, we do the same thing, and we get nxⁿ⁻¹. We keep applying this rule to this integral n. We apply it n times and we yield n(n-1)…2x. Notice that this coefficient is just our known factorial function, n!.
This may seem completely off topic, but it actually helps us a lot. Let us now think of the 0.5th derivative of x^(0.5). By our findings, we have that this is nothing but (0.5)!. Factorial of a number that’s not a natural number? That’s absurd! But what if I told you that we can extend this factorial function. Not just to positive rational numbers, but we can go even further. Matter of fact we can extend this onto complex numbers without the non-positive integers! We call this the Gamma function and it looks as follows:
