Question

Hello.
In lecture 6 of 18.01, when trying to prove what the derivative of a^x is, we use a "definition" that states "a^(x+y)=a^x+a^y". However this is in fact more a property than a definition and it comes out of nowhere. I thought the way to get every definitions right was first to define the logarithm function as the anti-derivative of 1/x, and then defining the exponential function as the inverse function of the logarithm. From there we can easily prove every of the properties of these two functions.
It seems to me that the proof presented is a tautology.

Answer 1

\[a^{(x+y)} \ne a^x + a^y\]

In the video they explain that..

\[a^{(x+y)} = \ a^x \cdot a^y \text{ or } a^xa^y \]
That's a^x times a^y.

We can see that this is true because of the definition of the exponent..

\(a^x = (a\cdot a \cdot\ ...\ \cdot a)\)
^ x of these
\(a^y = (a\cdot a \cdot\ ...\ \cdot a)\)
^ y of these

\(\implies a^x\cdot a^y = (a \cdot a \cdot \ ...\ \cdot a) = a^{x+y}\)
^ (x + y) of these

Answer 2

My bad, I made a typo. I meant \[a^{x+y}=a^xa^y \]
However my problem is precisely the definition of exponents. The one you state only works for x and y being positive integers. If we want to extend it to negative, we already have to make some changes in the initial definition and the same apply for rational numbers. So these are all extensions of the initial definition. Now we're stuck with irrational numbers, as our "already extended" definition which is: \[a^{x/y}=\sqrt[y]{a^x}\] doesn't work for all real (because of irrational numbers). So we want to extend one step further those definitions of exponentials. And to make quite sure you don't lose some of the initial property I think it is safer (and more formal) to redefine everything by going to way I described. Therefore I don't think it's a good idea to use something we want to define as a given definition.

Feel free to disagree and tell me why I am wrong :)

Answer 3

To derive the derivative of a^x, the method I would prefer is Taylor Series.
we have to evaluate lim y->0 (a^x*a^y-a^x)/y
now take a^x common from the numerator and then just expand the a^y using taylor series and ignore the parts which have y^2 and higher degrees.


Hope it helps! :)

Answer 4

Thanks for answering, but my question wasn't about how to differentiate the exponential function, but I was rather questioning the way it was taught in the video. I feel we use a property/definition/whatever which is derived from what we want to prove first, thus having a tautologic proof.

Answer 5

It wasn't derived. It was assumed to be known from high school algebra.

Answer 6

That's my problem, we assume we learned it. But did you prove that in high school? I bet you didn't, it was just being given as a definition/properties. Anyway, that doesn't really matter I guess, thanks for answering. I just find there is something a bit fishy about this way of presenting things.

Answer 7

The only part about which you can complain is when we make the exponential function include irrational exponents by the argument of continuity. It can be proven however so certainly feel welcome to convince yourself that it is the case regardless of the derivative.