In this explainer, we will learn how to find the derivatives of exponential functions.
Exponential functions play an important role in mathematics and have many physical applications. For example, we use exponential functions to model the growth of bacteria, continuously compounded interest, and radioactive decay, to mention just a few of its applications. In fact, its ubiquitous use in mathematics led one mathematician to call it โthe most important function in mathematics.โ One of the reasons why exponential functions are so important in mathematics is their properties in relation to derivatives; and in this explainer, we will explore these properties. We begin with a general definition of an exponential function.
Exponential Function
An exponential function is a function of the form where and are constants and .
A common misconception when it comes to derivatives of exponential functions is to assume that we can apply the power rule
This is not true as we shall see. It is important to be clear on the distinction: the power rule applies when the base of the exponent is the variable and the exponent is fixed, whereas for exponential functions the exponent is the variable and the base is fixed.
| The Power Rule | Exponential Functions |
|---|---|
The is the base. The exponent is . | The base is . The is the exponent. |
We will use the definition of the derivative to try to derive a formula for the derivative of an exponential function of the form . By definition, hence,
Using the rules of exponential functions, we can rewrite this as
We can now factor out as follows:
Since does not depend on , we have
Therefore, if the limit exists, this formula tells us that the derivative of an exponential function is proportional to the function itself.
Even more, we can say the expression is actually the definition of the derivative of at ; that is,
Therefore, if is differentiable at , it is differentiable everywhere and .
Numerically, we can approximate the value of this limit for particular values of . The table below includes numerical approximations of the limit.
| 0.1 | ||
| 0.01 | ||
| 0.001 | ||
| 0.0001 |
The numerical evidence certainly appears to suggest that this limit exists and, in fact, it can been proved that these limits exist.
Since the derivative of at is less than one and the derivative of at is greater than one, it seems reasonable to conjecture that there is a special value for such that its derivative at is one and, consequently, . This value does, in fact, exist, and mathematicians denote this number (Eulerโs number) and it is uniquely defined as the number that satisfies
The importance of this particular exponential function cannot be overstated. It is the unique exponential function that has a derivative of one at and its derivative is equal to its value for all . Furthermore, we will see that all exponential functions can be expressed in terms of this exponential function. It is often referred to as the natural exponential function or simply the exponential function.
Derivative of the Exponential Function
The derivative of the exponential function is
This formula can be generalized to
We will now look at some examples where we apply this formula to find the derivative of certain functions.
Example 1: Derivative of Exponential Functions
If , find .
Answer
Using the rule for differentiating exponential functions, we have
Example 2: Differentiating Functions Involving Exponentials
Differentiate the function .
Answer
Since the derivative of a sum is the sum of the derivatives, we can differentiate each term independently. Using the rule for differentiating exponential functions, we have and using the power rule, we have
Therefore, combining these two expressions, we have
Example 3: Differentiating Exponential Functions
Find if .
Answer
On first impression, it appears we have been given a function that is defined implicitly. However, since is nonzero, we can divide both sides of the equation through by which gives
Furthermore, we can divide through by 5 and rewrite as . Hence, which is a function defined explicitly. At this point, we can differentiate the function using the rule for differentiating exponential functions, to get
This can actually be expressed in terms of as
Using what we know about the derivative of the natural exponential function, we can find the derivatives of general exponential functions as we will demonstrate in the next couple of examples.
Example 4: Differentiating General Exponential Functions
If , determine .
Answer
Using the rules of exponentials and logarithms, we can write
Raising both sides to the power of , we have
Hence, we can rewrite the expression for as follows:
We can now use the rule for differentiating the natural exponential function, to get
We can convert our expression in terms of back to an exponent of base 2 to get
The following example demonstrated the rule for differentiating general exponential functions.
Derivative of General Exponential Functions
The derivative of an exponential function of the form is
This formula can be generalized to
Interestingly, we are now able to give an exact representation for the constants that we approximated numerically at the beginning of the explainer. In particular, which implies
Or, more generally,
Example 5: Differentiating General Exponential Functions
Find the first derivative of the function .
Answer
Although this expression looks complex, we can use the rules of exponents to significantly simplify the expression to get something we can easily differentiate. Using the power of powers rule, we can rewrite the expression for as
At this point, we might wonder how we can handle the fact that we have a binomial in the exponent. However, we can use the multiplicative rule of exponents, to rewrite this as
We now have an expression we can easily differentiate. Using the rule for differentiating general exponential functions, we have
We can recombine the product of exponential functions to rewrite this as
In our final example, we will consider a word problem involving differentiating exponential functions.
Example 6: Derivatives of Exponential Functions in a Real-World Context
The radioactive decay of radon-222 is modeled by the following formula: where is the remaining quantity, in grams, of radon-222 that has not decayed after days, is the initial quantity of radon-222, and is its half-life.
A particular sample initially contained 10 grams of radon-222. Given that the half-life of radon-222 is , find the rate of decay of the sample 10 days later. Give your answer to three significant figures.
Answer
When we are asked to find the rate of something, we should immediately think derivative. We have been asked for the rate of decay; therefore, we need to differentiate the function with respect to . Using the rule for differentiating general logarithms, we have
Using the laws of logarithms, we can rewrite this as
We need the rate of decay when ; therefore, substituting this in, along with and , we have
Key Points
- The derivative of the exponential function is which can be generalized to
- Similarly, the derivative of a general exponential function of the form is This formula can also be generalized to
- Before differentiating, it can be useful to simplify or manipulate the expression using the rules of exponents.
- Remember not to confuse the power rule and the rule for differentiating exponential functions.
The Power Rule Derivatives of Exponential Functions
The is the base.
The exponent is .
The base is .
The is the exponent.