Finding Derivatives of Exponential Functions w/ Step-by-Step Examples

An exponential function is a function of the form $f (x) = a^{x}$ , where $a$ is a positive constant not equal to 1, and $x$ is a variable. The derivative of an exponential function describes the rate of change of the function with respect to its variable.

Formula and Formal Definition

The derivative of an exponential function $f (x) = a^{x}$ , where $a > 0$ and $a \neq 1$ , is given by:

f^{'} (x) = a^{x} \ln (a)

For an exponential function $f (x) = a^{x}$ , where $a > 0$ and $a \neq 1$ , the derivative with respect to $x$ is:

\frac{d}{d x} (a^{x}) = a^{x} \ln (a)

where $\ln (a)$ is the natural logarithm of $a$ .

Derivative of the Exponential Function 𝑒ˣ

A special case of the exponential function is $f (x) = e^{x}$ , where $e$ is Euler’s number (approximately 2.71828). The derivative of $e^{x}$ is particularly simple:

f^{'} (x) = e^{x}

This is because $\ln (e) = 1$ , so:

\frac{d}{d x} (e^{x}) = e^{x} \ln (e) = e^{x} \cdot 1 = e^{x}

The Unique Property of the Exponential Function 𝑒ˣ

The exponential function $f (x) = e^{x}$ has a unique property: it is its own derivative! This means that the rate of change of $e^{x}$ is equal to the function itself. In other words, the slope of the tangent line to the graph of $e^{x}$ at any point is equal to the value of $e^{x}$ at that point.

This property makes the exponential function, $e^{x}$ , really special. In fact, it’s so special that it’s one of the most important functions in calculus and its applications. It’s the only function (up to a constant multiple) that has this amazing property! The exponential function $e^{x}$ pops up in lots of mathematical models of natural phenomena, such as population growth, radioactive decay, and compound interest. Why? Because its rate of change is directly proportional to its current value.

The unique property of $e^{x}$ can be derived from the definition of the exponential function and the properties of logarithms. Consider the function $f (x) = a^{x}$ , where $a > 0$ and $a \neq 1$ . If we set $a = e^{\ln (a)}$ , we can rewrite the function as:

f (x) = (e^{\ln (a)})^{x} = e^{x \ln (a)}

Using the chain rule, we can find the derivative of this function:

f^{'} (x) = e^{x \ln (a)} \cdot \ln (a)

If we set $a = e$ , then $\ln (a) = \ln (e) = 1$ , and we get:

f^{'} (x) = e^{x \ln (e)} \cdot \ln (e) = e^{x} \cdot 1 = e^{x}

Thus, the exponential function $f (x) = e^{x}$ is the only exponential function (up to a constant multiple) that is its own derivative.

Proof of the Derivative of an Exponential Function

To prove the derivative of an exponential function $f (x) = a^{x}$ , we can use the definition of the derivative:

f^{'} (x) = {lim}_{h \to 0} \frac{f (x + h) - f (x)}{h}

Step 1: Substitute $f (x) = a^{x}$ into the definition of the derivative.

f^{'} (x) = {lim}_{h \to 0} \frac{a^{x + h} - a^{x}}{h}

Step 2: Rewrite $a^{x + h}$ as $a^{x} \cdot a^{h}$ .

f^{'} (x) = {lim}_{h \to 0} \frac{a^{x} \cdot a^{h} - a^{x}}{h}

Step 3: Factor out $a^{x}$ .

f^{'} (x) = a^{x} {lim}_{h \to 0} \frac{a^{h} - 1}{h}

Step 4: Recognize that ${lim}_{h \to 0} \frac{a^{h} - 1}{h} = \ln (a)$ .

f^{'} (x) = a^{x} \ln (a)

Thus, we have proven that the derivative of an exponential function $f (x) = a^{x}$ is $f^{'} (x) = a^{x} \ln (a)$ . This proof relies on the properties of exponentials and logarithms, as well as the definition of the derivative.

Graph of the Derivative of an Exponential Function

The graph of an exponential function $f (x) = a^{x}$ depends on the value of $a$ :

If $a > 1$ , the function is increasing and the graph grows rapidly.
If $0 < a < 1$ , the function is decreasing and the graph approaches the x-axis as x increases.

The graph of the derivative of an exponential function $f^{'} (x) = a^{x} \ln (a)$ is similar to the original function, but it is scaled by a factor of $\ln (a)$ :

If $a > 1$ , then $\ln (a) > 0$ , so the derivative is positive and the graph is increasing.
If $0 < a < 1$ , then $\ln (a) < 0$ , so the derivative is negative and the graph is decreasing.

Here are some examples of exponential functions and their derivatives:

$5^{x}$ in blue
$2^{x}$ in orange
${(\frac{1}{2})}^{x}$ in green

The graphs illustrate the relationship between an exponential function and its derivative, with the solid lines representing the original functions and the dashed lines their derivatives. The derivative of an exponential function is proportional to the original function, where the constant of proportionality is $\ln (a)$ . For $0 < a < 1$ , the function $a^{x}$ decays as $x$ increases, and its derivative, which also decays, has a negative value because $\ln (a)$ is negative. The result is that both the function and its derivative are decreasing functions. Conversely, for $a > 1$ , the function $a^{x}$ increases exponentially and its derivative, scaled by $\ln (a)$ , is positive and also increasing.

Another property of all exponential functions is that their graphs goes through the point (0, 1), as $a^{0} = 1$ .

Applications of the Derivative of Exponential Functions

The derivative of exponential functions has numerous applications in various fields:

Population growth: Exponential functions can model the growth of populations, and their derivatives describe the rate of change of the population over time.
Radioactive decay: The decay of radioactive substances follows an exponential law, and the derivative of this function gives the rate of decay at any given time.
Compound interest: The growth of an investment with compound interest can be modeled by an exponential function, and its derivative represents the instantaneous rate of change of the investment’s value.
Cooling and heating: Newton’s law of cooling states that the rate of change of the temperature of an object is proportional to the difference between its temperature and the ambient temperature, leading to exponential functions and their derivatives.
Chemical reactions: The rates of many chemical reactions are described by exponential functions, and their derivatives give the instantaneous rate of change of the reactant or product concentrations.

Examples

Find the derivative of $f (x) = 3^{x} + 3^{x^{2}}$ .
Using the sum rule and the chain rule, we get:
$f^{'} (x) = 3^{x} \ln (3) + 3^{x^{2}} \cdot 2 x \ln (3)$
Find the derivative of $f (x) = \frac{e^{x}}{1 + x}$ .
Using the quotient rule, we get:
$f^{'} (x) = \frac{e^{x} (1 + x) - e^{x}}{(1 + x)^{2}} = \frac{x e^{x}}{(1 + x)^{2}}$
The population of a city is growing exponentially according to the function $P (t) = 50000 \cdot {1.03}^{t}$ , where $t$ is the number of years since 2000. Find the rate of population growth in the year 2050.
First, we find the derivative of $P (t)$ :
$P^{'} (t) = 50000 \cdot {1.03}^{t} \ln (1.03)$
Then, we calculate the rate of growth in 2050 by evaluating $P^{'} (50)$ :
$P^{'} (20) = 50000 \cdot {1.03}^{50} \ln (1.03) \approx 6479 people per year$