Exercise 11

Let $f(x) = a^x$. The goal of this problem is to explore how the value of $a$ affects the derivative of $f(x)$, without assuming we know the rule for $\frac{d}{dx}[a^x]$ that we have stated and used in earlier work in this section.

1. Use the limit definition of the derivative to show that $$\begin{gathered} f'(x) = \lim_{h \to 0} \frac{a^x \cdot a^h - a^x}{h} \end{gathered}$$.

2. Explain why it is also true that $$\begin{gathered} f'(x) = a^x \cdot \lim_{h \to 0} \frac{a^h - 1}{h} \end{gathered}$$.

3. Use computing technology and small values of $h$ to estimate the value of $$\begin{gathered} L = \lim_{h \to 0} \frac{a^h - 1}{h} \end{gathered}$$ when $a = 2$. Do likewise when $a = 3$.

4. Note that it would be ideal if the value of the limit $L$ was $1$, for then $f$ would be a particularly special function: its derivative would be simply $a^x$, which would mean that its derivative is itself. By experimenting with different values of $a$ between $2$ and $3$, try to find a value for $a$ for which $$\begin{gathered} L = \lim_{h \to 0} \frac{a^h - 1}{h} = 1 \end{gathered}$$.

5. Compute $\ln(2)$ and $\ln(3)$. What does your work in (b) and (c) suggest is true about $\frac{d}{dx}[2^x]$ and $\frac{d}{dx}[3^x]$?

6. How do your investigations in (d) lead to a particularly important fact about the function $f(x) = e^x$?