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Section2.7Derivatives of Exponential Functions

Now that we understand how derivatives interact with products and quotients, we are able to compute derivatives of

  • polynomials,
  • rational functions, and
  • powers and roots of rational functions.

Notice that all of the above come from knowing  1  the derivative of \(x^n\) and applying linearity of derivatives and the product rule.

There is still one more “rule” that we need to complete our toolbox and that is the chain rule. However before we get there, we will add a few functions to our list of things we can differentiate  2 . The first of these is the exponential function.

Let \(a \gt 0\) and set \(f(x) = a^x\) — this is what is known as an exponential function. Let's see what happens when we try to compute the derivative of this function just using the definition of the derivative.

\begin{align*} \diff{f}{x} &= \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} = \lim_{h \to 0} \frac{a^{x+h} - a^x}{h} \\ &= \lim_{h \to 0} a^x \cdot \frac{a^{h} - 1}{h} = a^x \cdot \lim_{h \to 0} \frac{a^{h} - 1}{h} \end{align*}

Unfortunately we cannot complete this computation because we cannot evaluate the last limit directly. For the moment, let us assume this limit exists and name it

\begin{align*} C(a) &= \lim_{h \to 0} \frac{a^{h} - 1}{h} \end{align*}

It depends only on \(a\) and is completely independent of \(x\text{.}\) Using this notation (which we will quickly improve upon below), our desired derivative is now

\begin{align*} \diff{}{x}a^x &= C(a)\cdot a^x. \end{align*}

Thus the derivative of \(a^x\) is \(a^x\) multiplied by some constant — i.e. the function \(a^x\) is nearly unchanged by differentiating. If we can tune \(a\) so that \(C(a) = 1\) then the derivative would just be the original function! This turns out to be very useful.

To try finding an \(a\) that obeys \(C(a)=1\text{,}\) let us investigate how \(C(a)\) changes with \(a\text{.}\) Unfortunately (though this fact is not at all obvious) there is no way to write \(C(a)\) as a finite combination of any of the functions we have examined so far  3  . To get started, we'll try to guess \(C(a)\text{,}\) for a few values of \(a\text{,}\) by plugging in some small values of \(h\text{.}\)

We can learn a lot more about \(C(a)\text{,}\) and, in particular, confirm the guesses that we made in the last example, by making use of logarithms — this would be a good time for you to review them.