In the field of Biology, it is common to study populations of bacteria through a microscope. Bacteria are known to have dependably regular replication intervals. Even when starting with only a small number of bacteria, the population can grow incredibly huge after only a short amount of time!
A population of Escherichia Coli, pixabay.com
At first, when the population is small, there will only be a few replications. However, the newly replicated bacteria will also begin replicating as well. Now, the population is growing at a faster rate! How can Calculus be used to describe this phenomenon?
The derivative of a function can also be seen as its Rate of Change. So the examples above can be thought of as functions whose derivatives are directly proportional to themselves! Let's explore this type of exponential function and its derivative in further detail.
Formulas for Derivatives of Exponential Functions
Let's look at the two cases for the derivatives of exponential functions: when the base is the number \(e \), and when it is not. If the base is \(e \)then we have a natural exponential function.
Derivative of the Natural Exponential Function
The natural exponential function has a very peculiar characteristic: it is its own derivative! Pretty cool right?
The derivative of the natural exponential function is the natural exponential function. In other words,
$$\frac{\mathrm{d}}{\mathrm{d}x}e^x=e^x.$$
Finding the derivative of an exponential function is pretty straightforward. Just keep in mind that you also have to use Differentiation Rules according to specific problems.
Graph of the Derivative of the Exponential Function
You found that the derivative of the natural exponential function is itself. This means that in the natural exponential function, the slope \( m \) of the line tangent to every point is equal to its y-value!
Fig. 1. Graph of the Exponential function with a tangent line at \(x=1\).
Derivative of Any Exponential Function
But what happens if the exponential function has a base other than \( e \)? Then you multiply by the natural logarithm of that base!
If the base \( b \) of the exponential function is other than \( e\), then:
$$\frac{\mathrm{d}}{\mathrm{d}x}b^x=(\ln b)b^x,$$
where \( b>0.\)
The above derivative is considered to be more general because if you let \( b=e \), then \(\ln{e}=1\), you get back to the first rule. So this one covers the case when the base is \( e\) as well as the case when the base is lots of other numbers.
You can use these differentiation rules along with rules like The Chain Rule and The Product Rule to find the derivatives of functions involving exponential functions.
Examples of Derivatives of Exponential Functions
Let's take a look at some examples of derivatives involving Exponential Functions.
Examples Using the Chain Rule
The Chain Rule can be used when finding the derivative of an Exponential Function.
Find the derivative of \(f(x)=e^{3x}.\)
Answer:
To find this derivative you will need to use the Chain Rule. Let \(u=3x\). Then by the Power Rule, \(u'(x) = 3 \). So now using the Chain Rule,
$$\begin{align} f'(x) &= \left(\frac{\mathrm{d}}{\mathrm{d}u} e^u \right)\left(\frac{\mathrm{d}u}{\mathrm{d}x}\right) \\ &= (e^u )(3) \\ &= 3e^{3x}, \end{align}$$
where you have used the fact that the derivative of the natural exponential function is just the natural exponential function. So \(f'(x) = 3e^{3x}\).
Similarly, you can find the derivative of more complex functions.
Find the derivative of \(g(x)=e^{x^2}.\)
Answer:
You will also need to use the Chain Rule here! This time, let \( u=x^2 \). Using the Power Rule you can find that \( u'(x)=2x \). You can now find the derivative of \(g(x)\) by working in a way similar to the previous example:
$$\begin{align} g'(x) &= \left( \frac{\mathrm{d}}{\mathrm{d}u} e^u \right) \left( \frac{\mathrm{d}u}{\mathrm{d}x} \right) \\ &= (e^u)(2x) \\&= 2xe^{x^2}. \end{align}$$
Examples Using the Product Rule and the Quotient Rule
Let's begin by looking at an example using The Product Rule.
Find the derivative of \( f(x) = x^2 e^x.\)
Answer:
Since this functions it the product of two functions, you can find its derivative by using The Product Rule,
$$\begin{align} \frac{\mathrm{d}f}{\mathrm{d}x} &= \left( e^x \right) \left( \frac{\mathrm{d}}{\mathrm{d}x}x^2 \right) + \left( x^2 \right) \left( \frac{\mathrm{d}}{\mathrm{d}x}e^x \right) \\ &= (e^x)(2x)+(x^2)(e^x) \\ &=e^x(x^2+2x). \end{align}$$
Let's now take a look at an example using The Quotient Rule.
Find the derivative of \(g(x)=\frac{e^x}{x+1} .\)
Answer:
This function now involves a quotient of functions, so you need to use The Quotient Rule,
$$\begin{align} \frac{\mathrm{d}g}{\mathrm{d}x} &= \frac{(x+1)\left(\frac{\mathrm{d}}{\mathrm{d}x}e^x\right) -(e^x) \left( \frac{\mathrm{d}}{\mathrm{d}x}(x+1) \right)}{(x+1)^2} \\ &= \frac{(x+1)(e^x)-(e^x)(1)}{(x+1)^2} \\ &= \frac{xe^x}{(x+1)^2}, \end{align}$$
where you have used the Power Rule to find the derivative of \(x+1.\)
Common Mistakes When Differentiating the Exponential Function
You can't be entirely literally when you say that the derivative of an exponential function is itself. This only applies to the natural exponential function. Let's see an example.
Find the derivative of \(f(x)=e^{x^2+x}.\) One common mistake is forgetting to use the Chain Rule:
$$\frac{\mathrm{d}f}{\mathrm{d}x}\neq e^{x^2+x}.$$
When you take the derivative using the Chain Rule you get
$$\dfrac{\mathrm{d}f}{\mathrm{d}x}=(2x+1)e^{x^2+x}.$$
Another mistake is using the Power Rule to differentiate the exponential function.
If you look at an exponential function, you might think of using the Power Rule to find the derivative of the function. However, this is a mistake!
$$\frac{\mathrm{d}}{\mathrm{d}x}3^{2x}\neq (2x)3^{2x-1}$$
You use The Power Rule when the variable is the base of the exponential expression. However, if the variable is the exponent, we need to use the differentiation rule for the exponential function. Also, don't forget to use the Chain Rule!
$$\frac{\mathrm{d}}{\mathrm{d}x}3^{2x}=2\left(\ln{3}\right)3^{2x}$$
Do not forget to multiply by the natural logarithm of the base when the base is other than \( e \)!
Find the derivative of \(5^{2x}.\) The base of this exponential function is 5, so this is not a natural exponential function. Another common mistake is forgetting to multiply by the natural logarithm of the base:
$$\frac{\mathrm{d}}{\mathrm{d}x}5^{2x}\neq (2) 5^{2x}.$$
Because the base of this exponential function is other than \(e\), you also have to multiply by the natural logarithm of the base to get
$$\frac{\mathrm{d}}{\mathrm{d}x}5^{2x}=2 \left( \ln{5} \right) 5^{2x}.$$
Definition of the Derivative of Exponential Functions
Remember that the definition of the derivative of a function involves the limit of the Rate of Change. In other words,
$$\frac{\mathrm{d}}{\mathrm{d}x}e^x=\lim_{h\rightarrow 0} \frac{e^{x+h}-e^x}{h}$$
You can rewrite the limit using the properties of exponents to factor out \(e^x\), giving you
$$\begin{align} \frac{\mathrm{d}}{\mathrm{d}x}e^x &=\lim_{h\rightarrow 0} \frac{e^x e^h - e^x}{h} \\&= \lim_{h\rightarrow 0}\frac{e^x \left( e^h-1 \right)}{h}. \end{align}$$
Since \(e^x\) does not depend on \( h, \) it can be removed from the limit, which gives you
$$\frac{\mathrm{d}}{\mathrm{d}x}e^x= e^x \lim_{h\rightarrow 0} \frac{e^h - 1}{h}.$$
Then taking the limit,
$$\frac{\mathrm{d}}{\mathrm{d}x}e^x=e^x.$$
You may wonder why
$$\lim_{h\rightarrow 0}\frac{e^h - 1}{h} = 1.$$
To find out, read on to the next section!
Proof of the Derivatives of Exponential Functions
You can use the limit definition to do a more formal proof of the derivative of the exponential function.
You have used the value of a limit in our proof of the derivative of the natural Exponential Function. Finding the value of this limit can be a little tricky, so let's take a careful look at the calculation.
Here you will find the value of the limit:
$$L=\lim_{h\rightarrow 0} \frac{e^h-1}{h}.$$
Begin by making the following substitution:
$$u=e^h-1$$
Note that \(u \rightarrow 0 \) as \(h \rightarrow 0.\) You can now rewrite the limit in terms of \( u \), which gives you
$$L=\lim_{u\rightarrow 0} \frac{u}{\ln{(u+1)}}.$$
You can also write the \(u \) in the numerator inside the limit as \( \frac{1}{u} \) in the denominator. Furthermore, you can rewrite this expression as the exponent of the natural logarithm using the Power Property of Logarithms. (See Properties of logarithms.)
$$L=\lim_{u \rightarrow 0} \frac{1}{\frac{1}{u} \ln{(u+1)}} = \lim_{u \rightarrow 0}\frac{1}{\ln{(u+1)}^{^1 / _u}}$$
Next, use the properties of limits to rewrite the limit inside the natural logarithm. You can do this because the natural logarithm is a continuous function:
$$L=\frac{1}{\ln{\left( \lim_{u\rightarrow 0}(u+1)^{^1 / _u} \right)}}.$$
The resulting limit within the denominator is the definition of the natural logarithm's base, \( e \), so
$$ L = \frac{1}{\ln{e}}.$$
Since \( e \) is the base of the natural logarithm, we know that \( \ln{e}=1. \) You have proved the required limit, and $L=1$ !
Proof of the Derivative for a General Exponential Function
The proof for the case when the base is not \( e \) relies on the fact that the exponential function and the natural logarithm function are inverses. This means that \( e^{\ln{a}}=a.\) You use this to your advantage!
$$\frac{\mathrm{d}}{\mathrm{d}x} b^x = \frac{\mathrm{d}}{\mathrm{d}x} e^{\ln{b^x}}$$
You can now use the power property of logarithms to rewrite the above expression as
$$\frac{\mathrm{d}}{\mathrm{d}x} b^x = \frac{\mathrm{d}}{\mathrm{d}x} e^{x\, \ln{b}}.$$
Next, you need to use The Chain Rule and the differentiation rule of the exponential function, which you have already proved above, to get
$$\frac{\mathrm{d}}{\mathrm{d}x} b^x = \left( \ln{b} \right)e^{x\ln{b}}.$$
Finally, undo the change you made in the first step. So \(e^{x\ln{b}}=b^x\), which means
$$\frac{\mathrm{d}}{\mathrm{d}x} b^x=\left(\ln{b}\right)b^x.$$
Derivatives of Exponential and Logarithmic Functions
Here you have looked at the derivative of both the natural exponential function and more general exponential functions. Since exponential and logarithmic functions are related, you can also take a look at the Derivative of Logarithmic Functions to see how their derivatives are related.
Derivative of the Exponential Function - Key takeaways
- The derivative of the natural exponential function is the natural exponential function. That is, $$ \frac{\mathrm{d}}{\mathrm{d}x} e^x = e^x.$$
- If the base is other than \( e, \) then: $$ \frac{\mathrm{d}}{\mathrm{d}x} b^x = \left( \ln{b} \right) b^x.$$
- The differentiation rule of the exponential function can be used alongside the chain rule, the product rule, and the quotient rule to find the derivative of any complex exponential function.
- The proof of the derivative of the exponential function can be done using limits.
- An important limit used in the proof of the derivative of the exponential function is: $$\lim_{h \rightarrow 0} \frac{e^h - 1}{h}=1.$$
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