Derivatives of Exponential Functions

Exponential Functions

We will now look at a theorem which will give us a rule for differentiating exponential functions.

Theorem 1: If $f$ is an exponential function such that $f(x) = a^x$ and $a > 0$, then the derivative $f'(x) = a^x \ln a$.

• Proof: The proof of this property relies on the Derivative Chain Rule and understanding that $\frac{d}{dx} \ln x = \frac{1}{x}$, both of which topics come up subsequently on Math Online. It is recommended that you come back to this proof later for verification.

• Let $y = a^x$ for some positive number $a > 0$, and take the natural logarithm of both sides to this equation to get that $\ln y = \ln (a^x) = x \ln a$. We note that since $\ln a$ is just a number, then:

(1)

\begin{align} \frac{d}{dx} \ln (f(x)) = \ln a \end{align}

• Now by the chain rule, we note that $\frac{d}{dx} \ln y = y' \cdot \frac{1}{y}$, and furthermore:

(2)

\begin{align} \frac{d}{dx} \ln y = y' \cdot \frac{1}{y} \\ \ln a = \frac{y'}{y} \\ y' = y \ln a \\ y' = a^x \ln a \quad \blacksquare \end{align}

Before we go further, we should look at a very special case where $a = e$ ($e \approx 2.71828...$):

Corollary 1 (The Derivative of $e^x$): If $f$ is an exponential function that has base $e \approx 2.71828$ (Euler's number), then $\frac{d}{dx} e^x = e^x$.

• Proof: Using theorem 1, we know that for $e = a > 0$ and so $\frac{d}{dx} e^x = \ln e \cdot e^x = e^x$ since $\ln e = 1$. $\blacksquare$

We will now look at some examples of differentiating exponential functions.

Example 1

Find the derivative of the following function:

(3)

\begin{equation} y = 3^x \end{equation}

The answer is rather simple:

(4)

\begin{align} \frac{d}{dx} 3^x = 3^x ln(3) \end{align}

The Exponential Function with Euler's Number as the Base

We will now turn out attention back to the function $f(x) = e^x$. This function is a very important function in calculus. It is a function that is defined as Euler's number, $e$ which is approximately equal to $2.71828...$.

(5)

\begin{equation} y = e^x \end{equation}

The interesting fact is that the derivative of this function is equivalent to itself:

(6)

\begin{align} \frac{d}{dx} e^x = e^x \end{align}

This implies that any point on the curve of $e^x$ is equivalent to that of the slope of the curve.