An exponential function is a function of the form:
\[\displaystyle a^{x}\]
where \(a\) is a non-zero number and \(x\) is a variable.
One should be careful and make the distinction between an exponential function such as \(2^x\) versus a polynomial such as \(x^2\) which is a variable to a numeric power/exponent.
If we are given \(f(x) = a^{g(x)}\) where \(g(x)\) is a different function of \(f(x)\). The derivative of \(f(x)\) is:
\[\displaystyle \begin{array} {lcl} f'(x) & = & \text{ln}(a) \cdot a^{g(x)} \cdot \dfrac{d}{dx} g(x) \\ & = & \text{ln}(a) \cdot a^{g(x)} \cdot g'(x) \end{array} \\ \]
Note that this general formula does use a variation of the chain rule. Since the exponent is a function of x, we take the derivative of the exponent as well.
Given the more common case of \(g(x) = x\) in the exponent, the general case becomes:
\[\displaystyle \begin{array} {lcl} f'(x) & = & \text{ln}(a) \cdot a^{x} \cdot \dfrac{d}{dx} x \\ & = & \text{ln}(a) \cdot a^{x} \cdot (x)' \\ & = & \text{ln}(a) \cdot a^{x} \cdot 1 \\ & = & \text{ln}(a) \cdot a^{x} \\ \end{array} \\ \]
If we are given \(e\) as the base such that we have \(f(x) = \text{e}^{g(x)}\). The derivative \(f'(x)\) will be as follows:
\[\displaystyle \begin{array} {lcl} f'(x) & = & \text{ln}(a) \cdot \text{e}^{g(x)} \cdot \dfrac{d}{dx} g(x) \\ & = & \text{ln}(e) \cdot \text{e}^{x} \cdot g'(x) \\ & = & 1 \cdot \text{e}^{x} \cdot g'(x) \\ & = & \text{e}^{x} \cdot g'(x) \\ \end{array} \\ \]
(Note that ln(e) = 1 as \(e^{1} = e\).)
A special case is where \(f(x) = \text{e}^{x}\)
In this case we have \(a = e\) and \(g(x) = x\). The derivative of \(\text{e}^{x}\) is simply \(\text{e}^{x}\).
Example One
Given \(f(x) = \text{e}^{2x}\), what is \(f'(x)\)?
\[\displaystyle \begin{array} {lcl} f'(x) & = & \text{ln(e)} \cdot \text{e}^{2x} \cdot \dfrac{d}{dx} 2x \\ & = & 1 \cdot \text{e}^{2x} \cdot 2 \\ & = & 2 \cdot \text{e}^{2x} \\ \end{array} \]
Example Two
\(f(x) = 2 \cdot \text{e}^{x}\)
\[ \displaystyle \begin{array} {lcl} f'(x) & = & 2 \cdot \dfrac{d}{dx} \text{e}^{x} \\ & = & 2 \cdot \text{e}^{x} \\ \end{array}\]
Example Three
\(f(x) = 5^{x}\)
\[\displaystyle \begin{array} {lcl} f'(x) & = & \text{ln(5)} \cdot 5^{x} \cdot \dfrac{d}{dx} x \\ & = & \text{ln(5)} \cdot 5^{x} \cdot 1 \\ & = & \text{ln(5)} \cdot 5^{x} \\ \end{array} \]
Example Four
\(f(x) = 10^{\text{sin}(x)}\)
\[\displaystyle \begin{array} {lcl} f'(x) & = & \text{ln(10)} \cdot 10^{\text{sin}(x)} \cdot \dfrac{d}{dx} \text{sin}(x) \\ & = & \text{ln(10)} \cdot 10^{\text{sin}(x)} \cdot \text{cos}(x)\\ \end{array} \]