The derivative of $a^x$, where $a$ is a constant is $\dfrac{d}{dx}(a^x) = \boxed{a^x \cdot \ln a}$. Let’s examine why this is true.

Finding the Derivative of $a^x$:

We will start by rewriting $a^x$ as:

\[\begin{align*} a^x = (e^{\ln a})^x = e^{\ln a \cdot x}. \end{align*}\]

Now, recall the formula $\dfrac{d}{dx}(e^{nx}) = e^{nx} \cdot n$. In the context of our problem, $n = \ln a$. Applying this to our expression, we get:

\[\begin{align*} \dfrac{d}{dx}(e^{\ln a \cdot x}) = e^{\ln a \cdot x} \cdot \ln a = \boxed{a^x \cdot \ln a}. \end{align*}\]

In general, converting exponential functions into a form with $e$ as the base can be a helpful trick for differentiating. Now, let’s look at an example.

A Simple Example: $2^x$

For this section, we will be using $2^x$ as our example. To find the derivative of this, we will be following the steps of the above section, with the stipulation that $a=2$. We start by rewriting $2^x$ as follows:

\[\begin{align*} 2^x = (e^{\ln 2})^x = e^{\ln 2 \cdot x}. \end{align*}\]

Next, we will use the formula for the derivative of $e^{nx}$, which is shown in the previous section. As a reminder, $n = \ln 2$ for this example. When we apply this to our current expression, we find that:

\[\begin{align*} \dfrac{d}{dx}(e^{\ln 2 \cdot x}) = e^{\ln 2 \cdot x} \cdot \ln 2 = \boxed{2^x \cdot \ln 2}. \end{align*}\]