In this explainer, we will learn how to find second- and higher-order derivatives of a function including using differentiation rules.
The derivative of a function tells us information about the rate of change of the function. This is an incredibly useful concept and it is used in a lot of real-world problems such as population growth and optimization. However, we do not need to stop there, we can also ask the question, “how is the rate of change itself changing?” We can determine this by differentiating the first derivative; we call this the second derivative. In fact, we can keep differentiating to find higher-order derivatives.
These higher-order derivatives are also useful in many real-world problems including spring–mass systems and the two-body problem. There are two main notations for derivatives: Leibniz’s notation and prime notation (sometimes referred to as Lagrange’s notation). Let’s start by extending these notations to higher-order derivatives.
Definition: Higher-Order Derivatives
As long as each derivative exists, we can differentiate functions any number of times and we can represent the nth derivative of a function as or in two ways using prime notation as where the prime symbol appears times.
We can evaluate the higher-order derivatives of a function at a value of . For example, the second derivative evaluated at could be written as
In our first example, we will find the second derivative of a polynomial function.
Example 1: Finding the Second Derivative of a Polynomial Function
Given that , determine .
Answer
means we need to differentiate twice with respect to . is a polynomial in , and we can differentiate polynomials term by term using the power rule for differentiation: for , ,
We can start by differentiating once to find :
Next, to find , we need to differentiate this expression again:
Hence,
In our second example, we determine the value of the second derivative of a function at a given point.
Example 2: Finding the Value of the Second Derivative of a Function at a Given Point
Determine the value of the second derivative of the function at .
Answer
We find the second derivative of by differentiating function to obtain and then differentiating again to obtain . To do this, we will start by rewriting our expression:
We can then differentiate this expression by using the power rule for differentiation: for ,
Hence,
We can differentiate again to find :
Finally, the question wants us to find the value of the second derivative at the point ; this is when . We substitute this value into the expression:
Hence, the value of the second derivative of the function at the point is .
In our next example, we will find the second derivative of a rational function.
Example 3: Finding the Second Derivative of Rational Functions
Given that , determine .
Answer
To find , we need to differentiate with respect to twice. Since we are given a rational function, we can find the first derivative by using the quotient rule.
If and are differentiable functions, then
We have
Since these are both polynomials, both functions are differentiable and, hence, we can differentiate them by using the power rule for differentiation: for ,
This gives
Substituting our expressions for , , , and into the quotient rule gives
We need to differentiate this expression to find . Since this is a rational function, we will do this once again by using the quotient rule. This gives
To help us differentiate , we will distribute the exponent:
We can then differentiate both functions by using the power rule for differentiation:
Substituting our expressions for , , , and into the quotient rule gives
Hence,
In our next example, we will use knowledge of the higher-order derivatives of a function to determine the values of coefficients in the function.
Example 4: Finding the Unknown Coefficients in the Expression of a Function given the Values of the Second and Third Derivatives of the Function
Given that , , and , find and .
Answer
We want to determine the coefficients of both terms in our function . We are given information about the third derivative of , , and the second derivative, .
Let’s start by finding expressions for these higher-order derivatives. Remember, to find , we need to differentiate twice, and to find , we need to differentiate three times.
We can differentiate a polynomial by using the power rule for differentiation: for ,
Applying this, we have
Differentiating again, we have
Then, differentiating one final time, we get
Remember, , and we know this is equal to . This gives
We are also told ; this means when we evaluate the second derivative at , we should get . Hence,
Substituting into this equation gives
Therefore, we have shown and .
In our next example, we will discover a pattern to determine a higher-order derivative of a trigonometric function.
Example 5: Finding the 𝑛th Derivative of an Expression
Find by finding the first few derivatives and observing the pattern that occurs.
Answer
We want to find ; this is the fifty-first derivative of with respect to . We could find this expression by differentiating this function fifty-one times; however, we will see there is a pattern in the derivatives.
Let’s start by recalling the following derivative results for trigonometric functions:
So, this gives us the first derivative of :
We find the second derivative of by differentiating the first derivative with respect to :
Following the same method, we find that
Hence, the fourth derivative of the sine function is also the sine function. This means every four times we differentiate the sine function, we will get the sine function. Therefore, since 48 is a multiple of 4,
To find the fifty-first derivative, we need to differentiate this three further times:
Hence,
We may need to use derivative rules multiple times in order to find some higher-order derivatives as we will see in our next two examples.
Example 6: Finding the Third Derivative of Trigonometric Functions Using the Product Rule
Find the third derivative of the function .
Answer
To find the third derivative of a function, we need to differentiate it three times. Since we are given a product of two differentiable functions, we will find the first derivative by using the product rule:
If and are differentiable, then
We will set
We can differentiate each of these functions separately:
Substituting these expressions into the product rule gives us
We need to differentiate this expression again to find the second derivative:
We will need to use the product rule for differentiation again; this time
We can differentiate each function separately:
Substituting these expressions into the product rule gives us
Hence,
We need to differentiate this once more to find the third derivative:
Since we already know , we will rewrite this derivative:
Therefore,
Example 7: Finding the Second Derivative of Root Functions
Given , find .
Answer
To find , we need to differentiate with respect to twice. Since we are given a composition of differentiable functions, we could differentiate this using the chain rule; however, we will use the general power rule.
If is a differentiable function and , then
First, we rewrite our function:
We set and ; since is linear, will be the slope of this line. Hence, .
Substituting these expressions into the general power rule gives us
To find the second derivative, we need to differentiate this expression:
We can do this by using the general power rule; we set and , giving us
In our final example, we will use the chain rule to find the second derivative of a trigonometric function.
Example 8: Finding the Second Derivative of Trigonometric Functions Using the Chain Rule
Given that , determine .
Answer
To find , we need to differentiate with respect to twice. We will start by finding the first derivative by recalling the following trigonometric derivative result. For any constant ,
Setting and taking the constant factor of 4 outside of the derivative gives us
We need to differentiate this once more to find the second derivative. Since this is a composition of differentiable functions, we will differentiate this function by using the chain rule:
If is differentiable at and is differentiable at , then
Setting and gives us
To use the chain rule, we need to differentiate and using the power rule for differentiation, and for any constant ,
Substituting these into the chain rule formula, we get
Finally, we substitute to get
Let’s finish by recapping some basic points.
Key Points
- We can differentiate functions multiple times to find higher-order derivatives.
- The th derivative of a function can be written as
- ,
- ,
- , where the prime symbol appears times.
- We can represent evaluating the derivative of a function using two different notations. For example, for the second derivative at , we can write
- ,
- .
