Higher-Order Derivatives
Higher-order derivatives are obtained by differentiating a function \( f(x) \) more than once.
Given a function \( y=f(x) \), we can compute its first derivative, which measures how quickly the function changes with respect to the independent variable \( x \).
\[ y'=f'(x) \]
Because the derivative is itself a function of \( x \), it can be differentiated again.
The derivative of the first derivative is called the second derivative and is denoted by \( y'' \) or \( f''(x) \).
\[ y''= [ f'(x) ]' \]
The process does not stop there. The second derivative can also be differentiated. The derivative of the second derivative is called the third derivative and is denoted by \( y''' \) or \( f'''(x) \).
\[ y'''= [ f''(x) ]' \]
By continuing in the same way, we obtain higher-order derivatives. As long as the function remains differentiable, the process can be repeated indefinitely.
This gives the fourth derivative \( y^{(4)} \), the fifth derivative \( y^{(5)} \), the sixth derivative \( y^{(6)} \), and so on.
\[ y^{(4)}=[f'''(x)]' \\ y^{(5)}=[f^{(4)}(x)]' \\ y^{(6)}=[f^{(5)}(x)]' \\ \vdots \]
These are collectively known as higher-order derivatives. In general, the derivative of order \( n \), or the \( n \)-th derivative, is written as \( f^{(n)}(x) \) or \( y^{(n)} \).
Note. The first three derivatives are usually written using prime notation: \( y' \), \( y'' \), and \( y''' \). Starting with the fourth derivative, mathematicians typically indicate the order in parentheses: \( y^{(4)} \), \( y^{(5)} \), \( y^{(6)} \), and so on. This notation is more practical and much easier to read.
What Are Higher-Order Derivatives Used For?
Higher-order derivatives are an essential tool in the analysis of functions.
The first derivative tells us where a function is increasing or decreasing. The second derivative provides information about concavity and helps identify inflection points.
Higher-order derivatives go even further. They reveal increasingly detailed information about how a function behaves and are widely used in approximation techniques and series expansions.
For example, higher-order derivatives play a central role in Taylor series and Maclaurin series, where a function is approximated by a polynomial near a given point.
In short, higher-order derivatives extend the concept of the first derivative and provide a deeper understanding of the local behavior of a function.
A Practical Example
Consider the function
\[ f(x)=x^3 \]
Its first derivative is
\[ f'(x)=3x^2 \]
This derivative describes the rate of change of \( x^3 \) at every point in its domain.
Differentiating once more gives the second derivative:
\[ f''(x)= ( 3x^2 )' = 6x \]
The second derivative provides information about the curvature of the graph.
Since the second derivative is still a function, it can be differentiated again to obtain the third derivative:
\[ f'''(x)= ( 6x )' = 6 \]
The third derivative is a constant function. Differentiating one more time gives the fourth derivative:
\[ f^{(4)}(x)= ( 6 )' = 0 \]
The fourth derivative is the zero function. From this point onward, all higher-order derivatives remain zero:
\[ f^{(5)}(x)= ( 0 )' = 0 \\ f^{(6)}(x)= ( 0 )' = 0 \\ \vdots \]
Since the derivative of the zero function is still zero, every subsequent derivative is also equal to zero.
Example 2
Now consider the function
\[ f(x)=x^4 \]
Its successive derivatives are:
\[ f'(x)=4x^3 \]
\[ f''(x)=12x^2 \]
\[ f'''(x)=24x \]
\[ f^{(4)}(x)=24 \]
\[ f^{(5)}(x)=0 \]
Once again, all derivatives of higher order are equal to zero.
Note. This is not a coincidence. For any polynomial of degree \( n \), the \( n \)-th derivative is a constant and the \( (n+1) \)-th derivative is the zero function. Since each differentiation reduces the degree of a polynomial by one, the process must eventually reach a constant and then vanish. As a result, every polynomial has only a finite number of nonzero derivatives.
Example 3
Not all functions behave like polynomials. Some functions can be differentiated infinitely many times without the process ever ending. Such functions are called infinitely differentiable.
A classic example is the sine function:
\[ f(x)= \sin x \]
Let us compute its successive derivatives.
The first derivative is
\[ f'(x)= \cos x \]
The second derivative is
\[ f''(x)= - \sin x \]
The third derivative is
\[ f'''(x)= - \cos x \]
Differentiating once again gives
\[ f^{(4)}(x)= \sin x \]
Notice what happened: the fourth derivative is exactly the same as the original function. At this point, the cycle starts over.
\[ f^{(5)}(x)=\cos x \]
\[ f^{(6)}(x)=-\sin x \]
\[ f^{(7)}(x)=-\cos x \]
\[ f^{(8)}(x)=\sin x \]
The derivatives of the sine function therefore follow a cyclic pattern with period 4. Every four differentiations, the original function reappears.
More generally, for every integer \( n \geq 0 \), the following relation holds:
\[ f^{(n+4)}(x)=f^{(n)}(x) \]
Note. The same phenomenon occurs for the cosine function. Its derivatives never become zero. Instead, they repeat in a periodic cycle. This is one reason why sine and cosine are among the most important examples of infinitely differentiable functions.
In summary, higher-order derivatives can behave very differently depending on the function being studied. For polynomials, repeated differentiation eventually leads to the zero function. For trigonometric functions such as sine and cosine, the derivatives never vanish but instead repeat in a predictable cycle.
And so on.
