WARNING - This post contains calculus, which may be unsuitable for liberal arts majors (Hats off to you, Randall Munroe)
The function f(x) = e^x* is widely regarded as very, very special. This is because it’s derivative, or the rate of change of f(x) to x, is also e^x. The slope of the function at a given point is exactly equal to itself.
An admirable quality, but there are some more interesting ones out there. One of the important properties of a derivative, is you can apply it to a function more than once. You can find a function’s derivative, it’s second derivative (taking the derivative twice), it’s third derivative, and so on, any number of times you wish. Each is a function, representing some change in f(x) to x. A question worth asking is, what happens when you take all these derivatives and add them together?
For many functions, the answer is ‘not much’. The function f(x) = x^3, for example, has the following derivatives:
- First derivative - 3x^2
- Second derivative - 6x
- Third derivative - 6
- Fourth derivative - 0
- Fifth derivative - 0
- All derivatives after the third - 0
Summed together, these derivatives add up to the polynomial 3x^2 + 6x + 6, which does have one noteworthy property: Each term is the derivative of the preceding term’s derivative. This is because a derivative of a polynomial with only one term, such as x^3, 3x, 12x^10, or 1600x^23, will always be a single term polynomial with order one less than the original polynomial. This is true of all following derivatives. So none of the derivatives will have any like terms, and adding them together is the same thing as writing them all next to each other with + signs in between them. Not that interesting.
Other functions have more astounding properties. The trigonometric function sin(x), diverges between four possible functions. Why? Because sin(x) has the derivative cos(x), which has the derivative -sin(x), which has the derivative -cos(x), whose derivative loops back to sin(x). This means that summing them, you can get four possible values.
sin(x)
sin(x) + cos(x)
sin(x) + cos(x) - sin(x) = cos(x)
sin(x) + cos(x) - sin(x) - cos(x) = 0
So the infinite series created by adding all the derivatives of sin(x) together doesn’t converge to any particular value, similar to adding 1 and -1 over and over.
What functions form series that converge then? Well, one of the best examples, and my personal favorite, is e^(x/2), or e to the one half of x. While the derivative of e^x is just e^x, the derivative of e^(x times a constant c) is equal to that constant times e^(cx). In the case of e^(x/2), the constant is one half. So it’s derivatives are:
- (1/2)e^(x/2)
- (1/4)e^(x/2)
- (1/8)e^(x/2)
- (1/16)e^(x/2)
And they continue that way, the same term reappearing at half the size each time. When they’re added together, the e^(x/2) portion can be factored out, leaving e^(x/2) * [ 1/2 + 1/4 + 1/8 + 1/16 …….. + 1/(2^n) + …. ]
The remarkable part is that the addition of those fractions converges to one. One half plus one fourth plus one eighth plus one sixteenth and so on to infinity gets arbitrarily close to one. This leaves e^(x/2) as the only term remaining. So when you take all the derivatives of the function f(x) = e^(x/2) and add them together, you’re left with exactly the same function, e^(x/2), much the same as the derivative of e^x is still e^x.
*The ^ symbol represents exponentiation, 3^2 is 3 raised to the power of 2, or three squared
