I see many proofs online and they all seem overly complicated. What would be wrong with this?
x = exp(ln(x))
Take the derivative of both sides
1 = exp(ln(x)) * (d/dx)ln(x)
1 = x * (d/dx)ln(x)
1/x = (d/dx)ln(x)
Where did you prove the chain rule?
Yes, that is commonly known as an application of the “chain rule” property. You are assuming that the relevant properties of the function exp(x) are already established and that ln(x) is known or defined to be its inverse.
What you have there is not a proof but a re-arrangement. You have shown that
d/dx e^y = e^y dy/dx
is equivalent to
d/dx ln z = 1/z dz/dx
which you can see in one step by setting y = ln z
If you want a proof, you need to start from some statement that does not use the derivative properties of ln x or e^x.
Nothing wrong with what you’re doing as it is an application of differentiation of the inverse. The general rule is typically introduced first, for example for finding the derivative of square roots.
There are many different pedagogical approaches to exp() and ln(). You can define exponentiation (first for integer exponents, then for reciprocal-integer, then for rational, and then for real), and then define e in some way, and then say that exp(x) is equal to e^x. You can define exponentiation, and then say that e is the constant such that the derivative of e^x is equal to 1 at x = 0. You can define exp() as the function which is its own derivative. You can define exp() in terms of its power series. You can define ln() as the inverse function of exp(), or you can define it as the integral of the reciprocal function, or in terms of its power series (with care taken for the finite radius of convergence). Once you’ve decided on your definitions, then you can prove all of the rest of those as consequences of those definitions.
Starting from the derivative properties of e^x is perfectly valid if he’s defined e^x that way, or if he’s already proven that from whatever his definition is without using the derivative of ln(x).
FTR: I am not looking for a proof (read the subject), merely a method to find the derivative of ln(x) and yes we can assume that we know (d/dx)e^x = e^x; exp(ln(x)) = x and the chain rule.
Look up proofs of this online and you will see how complicated they are such as using implicit differentiation* or dummy variables, . As simple as this is, my question is more along the lines of why wouldn’t this work? What am I missing?
- I know implicit differentiation is not difficult but you must admit using it here would make the steps more difficult for a student to follow.
In that case, what you have is fine.
Calculus is often taught as the first subject in mathematics where we really try to build everything from basic first principles. (Limits and whatnot.) Of course, Newton and Liebniz didn’t have limits, and made some intuitive assumptions about the properties of common functions. So if doing it that way is good enough for them, it can be good enough for you.
But one additional purpose of teaching calculus (aside from learning calculus) is to teach students how to approach more advances math topics in a logically sound, proof-focused way. So doing the extra work of proving the derivatives without a circular argument can be pedagogically useful.
But it can be overdone. My first number theory class was nothing more than a proof writing class using number theory as a backdrop. What was lost was a lot of the “big picture” and conceptual pieces.
It’s fine modulo the assumption that log(x) is differentiable. It looks like you’re using the definition of log as in the inverse of exp, in which case it follows from the inverse function theorem. The same argument shows that (f[SUP]-1[/SUP])’(f(x)) = 1/f’(x) in general (and this is sometimes taken as part of the inverse function theorem), though again the trickier part is showing that the derivative exists.
Isn’t the role of a beginner’s mathematics course traditionally filled by something like Euclidean geometry? You don’t even have to know how to add and subtract for that.
Calculus seems a bit too specialized, and depending on the curriculum track it may be possible to graduate from high school without studying it at all.
The exponential function is generally defined in such a way that it is its own derivative. Generally, it is shown that the derivative of a^x is a multiple of a^x and then e is the unique number such that the constant is 1. Then the ln is the inverse function (this essentially goes all the way back to Briggs, who first created logarithms) and the OP is correct. Of course you need the chain rule, but that is fundamental to all of calculus.