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 an Austrian American mathematician, Walter Rudin, to call it the most important function in mathematics. One of the reasons why exponential functions are so important in mathematics is their properties in relation to derivatives; in this explainer, we will explore these properties. We begin with a general definition of an exponential function.
Definition: Exponential Function
An exponential function is a function of the form where is a constant satisfying and . is called the base of the exponential function.
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.
| Power Functions | Exponential Functions |
|---|---|
| The is the base. | The base is . |
| The exponent is . | The is the exponent. |
We can differentiate power functions by using the power rule. For any ,
However, the power rule can only apply to power functions and not to exponential functions. Application of the power rule to differentiate an exponential function is a common mistake in differentiation. For instance, applying the power rule to find the derivative of would result in , which is incorrect. We need to be careful to recognize that the variable in is in the exponent, which makes it an exponential function. Hence, we cannot apply the power rule to differentiate this exponential function.
Let us consider how to differentiate an exponential function. We will begin by finding the derivative of the natural exponential function, , whose base is known as Euler’s constant. Euler’s constant is defined through the following limit:
After finding the derivative of the natural exponential function, we will learn how to differentiate general exponential functions, which are in the form for some and .
Recall the derivative of a function is given by
Applying this definition to the natural exponential function gives us
Using the rule of exponents, we can write as . Then, this limit is written as
In the numerator of the quotient, we note that is a common factor. Hence, the factorized form is given by
Since does not depend on , we can treat it as a constant in regard to the limit as . Hence, we can factor out this expression from the limit to write
Hence, evaluating the limit will lead to the formula for the derivative of . Let us find this limit.
Computing this limit involves a few tricks, which are not apparent at first. But it will be helpful to see how exactly the definition of Euler’s constant given in (1) impacts the computation of this limit.
We begin by defining variable to equal the numerator of the quotient within the limit:
We can rearrange this equation so that is the subject:
We can substitute the numerator of the quotient by and the denominator of the quotient by . We also note that when approaches 0, the variable also approaches 0 since . Changing the variable of the limit from to , we can write
Multiplying the numerator and denominator by , we can write
Recall the law of logarithms that tells us that the coefficient of a log becomes the exponent of its argument. Hence, applying the concepts of continuous functions, we can write the limit above as
Now, consider the limit within the parenthesis above. This resembles somewhat the limit in equation (1), which is the definition of Euler’s constant. Let us define another variable by
This means also that . We note that as approaches 0, approaches infinity. This leads to
By equation (1), the limit on the right-hand side is equal to . Substituting the value of this limit in (3), we can find the value of the desired limit:
Now that we have found the value of this limit, we can return to differentiating . Substituting this limit value in equation (2), we have
This gives us the derivative of the natural exponential function.
Rule: Derivative of the Natural Exponential Function
The derivative of the natural exponential function is
The significance of the derivative rule for the natural exponential function cannot be overstated. The natural exponential function is the only nonzero function whose derivative is equal to itself. This fact makes the natural exponential function appear as a solution for many different mathematical models of real-world problems. Beyond any doubt, the natural exponential function is the most prominent function in mathematical models of the real world.
Let us begin with an example where we apply this rule to differentiate a given function.
Example 1: Differentiating Functions Involving Exponentials
Differentiate the function .
Answer
In this example, we need to differentiate a function that involves the exponential function and a root function . We note that the given root function can be expressed as a power function by
Using this form, we can rewrite the given function as
Hence, the derivative is written as
We recall the sum/difference rule that allows us to split up the sum inside the derivative and also the constant multiple rule that allows us to factor the constants 3 and 5 outside the derivatives. Hence,
Now, we need to differentiate the exponential and power functions. We recall the rules of differentiation for these functions:
Applying these rules to , we obtain
Hence,
In the next example, we will use the derivative of the natural exponential function multiplied to a reciprocal trigonometric function.
Example 2: Finding the Derivative of a Function Involving Trigonometric and Exponential Functions Using the Product Rule
Differentiate .
Answer
We can begin by noticing that the given function is a product of two functions, and . We recall the product rule: given differentiable functions and ,
Applying the product rule to our function,
Now, we need to compute the derivatives and . We recall the derivatives of the exponential and reciprocal trigonometric functions:
Substituting these expressions into , we have
We can see that is a common factor of both terms on the right-hand side of the equation. Factoring out this term, we have
In our next example, we will apply the derivative rule for the natural exponential function combined with the chain rule to differentiate a given function.
Example 3: Differentiating Exponential Functions Using the Chain Rule
If , find .
Answer
In this example, we need to find
Before we differentiate the exponential function, we can factor out the constant outside the derivative using the constant multiple rule:
Now, we consider the derivative . We can note that is a composition of two functions; hence, we recall the chain rule for differentiating a composition of functions:
For the function , the outside function is and the inside function is . We need to compute the derivatives of each of these functions. To compute the derivative of , we recall the rule for differentiating the exponential function,
This leads to
On the other hand, we can apply the power rule to differentiate
Substituting these expressions into the chain rule, we have
Finally, remembering that we have factored out in the beginning, we can write
In the previous example, we applied the chain rule to differentiate an exponential function. Using the same approach, we can write a more general formula for chain rules that involve the exponential function.
Rule: Chain Rule for the Natural Exponential Function
Given a differentiable function , we have
When we differentiated in the previous example, we could have noted that this is the chain rule with the exponential function where . Since , this leads to the derivative
This is not a formula that must be memorized since we can always find this derivative by writing out the chain rule as we did in the first example. However, knowing this extra rule will often save our time when we differentiate these functions.
In the next example, we will apply this rule to differentiate an exponential function with a quadratic exponent.
Example 4: Differentiating Natural Exponential Functions Using the Chain Rule
Determine the derivative of .
Answer
In this example, we need to find the derivative
We can factor out the constant outside the derivative using the constant multiple rule:
To compute the derivative , we note that this is a composition of functions in the form , where . Recall the chain rule for the natural exponential function:
Using the power rule, we can compute
Substituting this derivative into the chain rule, we have
Finally, remembering that we have factored out in the beginning, we can write
Hence,
In the next example, we will apply the chain rule for the natural exponential function with the quotient rule to differentiate a given function.
Example 5: Differentiating Combinations of Exponential and Polynomial Functions Using the Quotient Rule
Find the first derivative of the function .
Answer
We can begin by noticing that the given function is a quotient of two functions, and . Hence, we recall the quotient rule: given two differentiable functions and ,
Applying the quotient rule to our function, we have
We now need to compute the derivatives and . The derivative by applying the power rule. For the other derivative, we can first apply the constant multiple rule to factor out the constant 4 from the derivative:
To compute , we recall the chain rule for the natural exponential function:
In this case, ; hence, by the power rule. Substituting these expressions into the chain rule, which means
We can now substitute these derivatives into to obtain
So far, we have considered the rule for differentiating the natural exponential function. We now turn our attention to the general exponential function , where the base satisfies and . We can relate a general exponential function to the natural exponential function when we use the identity
If we apply this identity with and , then writing the log-based as (natural log),
Recall the property of logarithm that states . This leads to
Hence, a general exponential function can be written in form of a natural exponential function. To find the derivative of a general exponential function, we can apply the chain rule for the natural exponential function with . We can apply the power rule to obtain , so we can write
Since , we can substitute this expression to obtain the derivative of a general exponential function.
Rule: Derivative of General Exponential Functions
The derivative of an exponential function , if and , is
Let us consider an example where we use the derivative of a general exponential function to find the derivative of a given function.
Example 6: Finding the First Derivative of an Exponential Function with an Integer Base
If , determine .
Answer
In this example, we need to determine
We can factor out the constant outside the derivative using the constant multiple rule:
To compute the derivative of the exponential function , we recall the derivative of an exponential function:
Since we have the base for our exponential function, we can write
Substituting this derivative into , we can obtain
In our next example, we will apply the chain rule with the derivative of a general exponential function. While we have established a formula for the chain rule for the natural exponential function, this formula does not directly lead to the chain rule for general exponential functions. In an effort to avoid over-producing formulae, we will compute these derivatives by applying the chain rule directly.
Example 7: Differentiating Exponential Functions Using the Chain Rule
Find the first derivative of the function .
Answer
In this example, we need to find the first derivative of an exponential function that is raised to another exponent. Although this expression looks complex, we can use the rules of exponents to significantly simplify the expression. We recall that, for any base and exponents and ,
In other words, we can multiply these exponents together. Using this rule, we can simplify the given function as
Now, we compute the first derivative:
We can note that is a composition of two functions; hence, we recall the chain rule for differentiating a composition of functions:
For the function , the outside function is and the inside function is . We need to compute the derivatives of each of these functions. To compute the derivative of , we recall the rule for differentiating general exponential functions,
This leads to
On the other hand, we can apply the power rule to differentiate
Substituting these expressions into the chain rule, we have
In the previous example, we applied the chain rule for a general exponential function to differentiate a given function. We can follow the same process to find a general formula for the chain rule for general exponential functions.
Rule: Chain Rule for General Exponential Functions
Given a differentiable function and a constant satisfying and , we have
In our final example, we will consider a real-world problem involving the derivative of the natural exponential function.
Example 8: Finding the Rate of Change of Exponential Functions in a Real-World Context
A factory’s production of units in a day is governed by the relation . What is the rate of change of production with respect to time on the fifth day?
Answer
We recall that the rate of change of a function is given by the derivative of the function evaluated at a point. In this example, our function represents a factory’s production. To find the rate of change of production with respect to time on the fifth day, we need to find the derivative of this function and evaluate it at .
Let us find the derivative . We can apply the constant multiple rule and sum/difference rule to write
We now need to compute the derivatives and . By the constant rule, we know that
To compute the derivative , we recall the chain rule for the natural exponential function: given a differentiable function ,
Here, ; hence, we have by the power rule. Substituting these derivatives in the chain rule,
We can now substitute these derivatives into to write
Evaluating the derivative at ,
Hence, the rate of change of production with respect to time on the fifth day is .
Let us finish by recapping a few important concepts from this explainer.
Key Points
- The derivative of the natural exponential function is
- Given a differentiable function , we have
- The derivative of an exponential function , if and , is
- Given a differentiable function and a constant satisfying and , we have