Math Insight

Exploring the derivative of the exponential function

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  1. Let $f(x)=2^x$.
    1. Is it true that the derivative of $f(x)$ is $x \cdot 2^{x-1}$? Justify your answer.
    2. Calculate $(2^h-1)/h$ for increasingly smaller values of $h$ until you are confident that you have estimated $\lim_{h \to 0} (2^h-1)/h$ to four decimal places. Be sure to test both positive and negative $h$.
    3. Using the limit definition of the derivative, show that \begin{align*} f'(0) = \lim_{h \to 0} \frac{2^{h}-1}{h}. \end{align*}
    4. Using your estimate of the limit $\lim_{h \to 0} (2^h-1)/h$, write down a formula for $f'(x)$.

  2. Let $g(x)=b^x$, where the base $b$ is a positive parameter.
    1. Using the limit definition of the derivative, show that \begin{align*} g'(x) &=g(x) g'(0) \end{align*} where \begin{align*} g'(0) = \lim_{h \to 0} \frac{b^h-1}{h}. \end{align*}
    2. Pick two values of $b$ besides $b=2$. Try one value of $b$ less than one and one value of $b$ greater than one. For each of your choices for $b$, use the first applet to estimate the first four digits of $g'(0)$. Explain how you used the second applet to verify this result.

  3. Obtain an estimate of $e$ by finding values of $b$ where $\lim_{h \to 0} (b^h-1)/h$ is very close to one. Based on your calculations, how many digits of your estimate can you trust?

  4. Let $f(x)=ce^{kx}$ where $c$ and $k$ are parameters, and $e$ is the number that satisfies $\lim_{h \to 0} (e^h-1)/h = 1$ .
    1. How does the relationship between the derivative $f'(x)$ and the function $f(x)$ depend on the parameters $c$ and $k$? Explain how to determine this by using the second applet.
    2. What is the relationship between the limit $\lim_{h \to 0} (e^h-1)/h$ and the limit $\lim_{w \to 0} (e^w-1)/w$? Justify your answer.
    3. What is $f'(x)$?
    4. If $c=-17$ and $k=-5$, what is $f'(x)$?
    5. What is $\diff{}{x}\left(14e^{-x}\right)$?

    1. In question 1, you estimated $\lim_{h \to 0}(2^h-1)/h$. How many digits of your estimate matched $\ln 2$?
    2. Which functions are their own derivatives?
    3. Imagine you were given the “differential equation” $f'(x)=f(x)$. What functions $f(x)$ could satisfy this equation?
    4. Imagine you were given the “differential equation” $f'(x)=3f(x)$. What functions $f(x)$ could satisfy this equation?

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