So recently I was teaching high school calculus to a high school class and I was wondering about the pedagogically best way to make students actually understand why the derivatives of the exponential & logarithm functions are what they are (instead of just being told what they are). The students already knew some basic calculus including derivatives of power functions and rules of differentiation. The following is the approach that I came up with and I am curious to learn how it compares to alternative approaches and whether people think that those alternative approaches are of more pedagogical value, and why.

  1. Start by tabulating values and making precise drawings of $2^x$ and $3^x$, then draw tangents at $x=0$ and work out their slope with a ruler and calculator. You get approx. $0.7$ and $1.1$, so arguably there must be a value $2 << a < 3$ for which at $x=0$ the function $a^x$ has a tangent with slope exactly $1$. Call this value Euler's number $e$ (thereby establishing a working definition of $e$) and consider the corresponding exponential function $f(x)=e^x$ with $e$ as a base.

  2. From that definition of $e$ we know that $f'(0)=1$, which translates to $\lim\limits_{h\to 0}\frac{e^h-1}{h}=1$ by first principles. Rearranging this for $e$ gives $(1+h)^{\frac{1}{h}}\to e$ as $h\to 0$, from which students can work out the value of $e=2,718...$ with a calculator.

  3. Now work out the derivative of $e^x$ from first principles (differential quotient). This is easy enough, because $e^{x+h}=e^x e^h$ and the limit $\lim\limits_{h\to 0}\frac{e^h-1}{h}=1$ is known from the chosen definition of $e$. One finds that $f'(x)=e^x$, which is a cool result.

  4. Define the natural logarithm $\ln x$ as the inverse function of $e^x$ (logarithms and inverse functions are known). In order to work out the derivative of $\ln x$ start with the ansatz $e^{\ln x} = x$, which is clear from the chosen definition of $\ln x$. Differentiate both sides by applying the chain rule and using the now known fact that $e^x$ is its own derivative. This way one immediately obtains the result that $\ln x$ has derivative $\frac{1}{x}$.

  5. Given this, it is now easy to work out the derivative of general $a^x$ and $\log_a x$ by chain-ruling the expression $a^x = (e^{\ln a})^x = e^{\ln a\times x}$ and differentiating $\log_a x=\frac{\ln x}{\ln a}$. From this it becomes clear that the original exact slope values were $\ln 2$ and $\ln 3$ and could have been worked out by pressing a calculator button, which students found amusing.

In fact, this definition of $e$ can be made rigorous (assuming familiarity with uniform convergence) and thereby may be of some pedagogical value at undergraduate university level, too.

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    $\begingroup$ One thing I've done is to show how consecutive differences of $n$'th powers grow like $(n-1)$'th powers (arithmetic sequences have constant consecutive differences, quadratic functions of the positive integers have arithmetic sequences for their consecutive differences, etc. -- don't prove in general, just show for squares and cubes) whereas consecutive differences of $2^n,$ $3^n,$ etc. grow exponentially with the same base. I've written a lengthy post about this, but I can't find it, so it was probably in Math Forum's ap-calculus discussion group, whose archives appear to no longer exist. $\endgroup$ Sep 22 '21 at 16:11
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    $\begingroup$ Use several approaches! Each approach reviews different skills and connects different concepts together. Be sure to talk about exponential growth in one of them: if a population is constantly doubling, the rate it grows depends on how large it is. $\endgroup$
    – TomKern
    Sep 22 '21 at 20:11
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    $\begingroup$ This is essentially the approach I take in university calculus. My definition of $e$ is that it is the number for which $\lim_{h \rightarrow 0} \frac{e^h-1}{h} = 1$. Of course this can be motivated by exactly the graphs you mention. Admittedly, we use the inverse function theorem in these arguments. Nobody is proving that in Calculus I. So, Calculus 1 always has holes logically. Of course, there is also the logarithms via integral definition, but I find those calculations awkward and I really prefer to introduce exponentials and logs much earlier in the course. In short, you're good. $\endgroup$ Sep 23 '21 at 3:25
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    $\begingroup$ I don't see a question here, so voting to close. $\endgroup$ Sep 25 '21 at 23:45

I normally go other way around.

Start with a logarithm, which by definition maps multiplication to addition: $\ln{ab} = \ln{a} + \ln{b}$.

A detour into a history could be also useful: multiplication is hard, addition is easy; this mapping, they say, prolonged the lives of astronomers.

Prove, by a sheer geometrical argument, that the $\displaystyle \int_1^x\dfrac {1}{t}dt$ satisfy the above definition. You may shamelessly use the idea of an area of a curvilinear trapezoid. They will understand.

Use a fundamental theorem to demonstrate that $(\ln{x})' = \dfrac{1}{x}$

Introduce the inverse function, $f$. Show that it maps addition to multiplication. Show how it naturally extends the idea of power. Use the the inverse function derivative theorem to demonstrate that $f' = f$

Make sure to spend time to discuss how $e$ emerges, how other bases come to play, and what makes $e$ special.


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