Why do all elementary functions have an elementary derivative?

Question

Considering many elementary functions have an antiderivative which is not elementary, why does this type of thing not also happen in differential calculus?

Answer 1

Just think of how we find those elementary functions:

• We start with the constant functions, which have derivative $0$, and the identity function $f(x)=x$ which has derivative $1$.

• We combine functions by means of addition, subtraction, multiplication, division, composition. For all of those cases we have explicit rules for the derivative.

• We define new functions as the integral of other functions (e.g. $\ln x$ as integral of $1/x$). Obviously when deriving those we get back the function we started with.

• We define functions as the inverse of another function. Again, we've got an explicit formula for derivatives of inverse functions.

Any function that cannot be defined by a chain of such operations (and also some which can, using the integration rule) we don't consider elementary.

So basically the reason is in the way we construct elementary functions. In some sense, one could say it is because of what functions we consider elementary.

Indeed, this hold not only for elementary functions; even most non-elementary functions we use are defined through such operations (in particular by integrals).

Answer 2

The short answer is that we have differentiation rules for all the elementary functions, and we have differentiation rules for every way we can combine elementary functions (addition, multiplication, composition), where the derivative of a combination of two functions may be expressed using the functions, their derivatives and the different forms of combination.

Integration, on the other hand, neither has a direct rule for multiplication of two functions nor for composition of two functions. We can integrate the corresponding rules for differentiation and get something that looks like it (integration by parts and substitution), but it only works if you're lucky with what elementary functions are combined in what way.

You might say that there is a hope that there are rules out there, that we just haven't found them yet. This is not true; it's been proven that there are always integrals of elementary functions that are not elementary themselves (under most reasonable definitions of "elementary functions"). It's a deep result known as Liouville's theorem.

Answer 3

Elementary functions are linear combination or quotient of polynomial and exponential functions (yes, that include sin, cos, tan). No doubt polynomials has derivative to be polynomials. And exponential is the single most important function in the universe, and clearly it is always differentiable and produces only exponential functions. The whole process of taking derivative is a combination of above process, so they are certainly elementary.

Answer 4

$\exp$, $\sin$, $\cos$, $\log$... are solutions of very simple differential equations. This fact plus the existence of explicit rules for the derivatives of sum/difference/product/quotient/composition guarantee that no "new" functions will appear when deriving elementary functions.

Answer 5

We can prove this:

From Wikipedia, an elementary function is one of the types:

1. $\exp,\log,\operatorname{id}$
2. Constant
3. The sum of two elementary functions.
4. The product of two elementary functions.
5. The difference of two elementary functions.
6. The quotient of two elementary functions.
7. The composition of elementary functions.

We shall show that every case has an elementary derivative.

1. $\mathrm D\exp = \exp$, which is of category (1). $\mathrm D\log x = x^{-1}$, which are elementary, with $x^{-1}$ being of category (6) -> (2) -> (1)
2. The derivative of a constant is $0$, which is elementary of category (2).
3. Follows from the linearity of differentiation: $\mathrm D(f+g) = \mathrm Df + \mathrm Dg$. Under the assumption that $f,g$ has elementary derivatives, the derivative of $f+g$ is elementary.
4. Similar to (3), via the product rule.
5. (2),(3),(4)
6. Similar to (3), via the quotient rule.
7. Chain rule.

Since every elementary function must belong to one of the categories above, it must have an elementary derivative.

Answer 6

• for the same reason that there are strict rules for grammar, but not for creating poems. Differentiation goes ‘downwards’ – in the direction of gravity, so to speak – whereas integration goes ‘upwards’ – against the pull of gravity, so to speak. The price of defeating gravity is ambiguity, which takes the form of a lack of strict rules. Notice that integration gives you a taste of ambiguity from the very start: changing the value of a function a single point does not change the value of the integral.