Introduction to Higher order derivatives
The higher order derivative is the rate of change of a function that is calculated by differentiating it repeatedly. It is calculated by differentiating (n-1)th derivative of a function. The higher order derivative is used for different purposes such as the second derivative test allows us to determine the nature of the function. Let’s discuss how to compute higher order derivatives and the relation between higher derivative and product rule.
Understanding the Higher Order Derivatives
A derivative in calculus, is the calculation of rate of change in a function with the change in its variable. It is usually known as the first derivative of the given function and denoted by f’(x). The higher order derivative of a function is referred to the rate of change in f(x) up to n times. In other words, it includes the repeated differentiation of the functions such as calculating first, second, third and successive derivatives. By definition, the higher order derivative is defined as:
“The higher order derivative of a function can be calculated by differentiating the first derivative repeatedly up to n times.”
The nth order derivative of a function is significant because it helps to determine the behaviour of a function at every point in its domain.
Higher Order Derivative Formula
The higher order derivative is a process of differentiating a function repeatedly to obtain its first, second, third and (n-1)th derivative. Every derivative of the functions f(x) can be obtained by differentiating the previous derivative. The formula to differentiate a function n times is:
$\frac{d^n}{dx^n}[f(x)]=\frac{d}{dx}\left(\frac{d^{n-1}}{dx^{n-1}}f(x)\right)$/p>
Where,
- $\frac{d^n}{dx^n}$ represents the nth derivative.
- $\frac{d^{n-1}}{dx^{n-1}}$ represents the (n-1)th derivative.
For any value of n, we can determine the relation between the derivatives of a function. For example we have to calculate 6th derivative. So, we will substitute n=6 in the above formula.
$\frac{d^6}{dx^6}[f(x)]=\frac{d}{dx}\left(\frac{d^{6-1}}{dx^{6-1}}f(x)\right) = \frac{d}{dx}\left(\frac{d^5}{dx^5}f(x)\right)$
Hence to calculate 6th derivative of f(x), we have to differentiate the 5th derivative of f(x).
Higher order derivative and product rule
The product rule for higher derivatives is a formula to calculate higher derivative of a product of two functions. For example, to calculate higher-derivative of a function f(x)g(x), we have to use product rule which is expressed as;
$\frac{d^n}{dx^n}[f(x)g(x)] =\sum_{k=0}^n(n k) f^{n-k}g^{k}$
Where,
- $\frac{d^n}{dx^n}[f(x)g(x)]$ represents the product rule for higher order derivative.
- $\sum_{k=0}^n(n k) f^{n-k}g^{k}$ represents the product of derivatives of $f(x)g(x)$.
By using above formula, we can find the first, second and other successive derivatives of the product f(x)g(x). The list of some derivatives of f(x)g(x) by using above formula is,
Higher order derivative and quotient rule
The quotient rule for higher derivatives is a method to calculate derivative of a quotient of two functions. The formula to calculate higher derivative of a quotient f(x)/g(x) is,
$\frac{d^n}{dx^n}[f(x)g(x)]=\frac{g(x)f^n(x)-f(x)g^n(x)}{[g(x)]^2}$
Where,
- $f^n(x)$ is the nth derivative of f(x).
- $g^n(x)$ is the nth derivative of g(x).
But we cannot directly calculate the higher-derivative by using the quotient rule. For this, we required the product rule formula for higher derivative. For example the second derivative of a quotient of h=f/g is,
$''(x)=\frac{f''-g''h-2g'h'}{g}$
How to find higher order derivative?
The calculations of higher-order derivatives are divided into a few steps. These steps assist us to calculate the derivative of a function having more than one independent variable. These steps are:
- Write the expression of the function.
- Identify the number of independent variables.
- Differentiate the function with respect to x. It will give you the first derivative.
- Differentiate the first derivative again to get the second derivative and so on.
- Simplify if needed.
Let’s understand the following examples to find the nth derivative.
Higher Order Derivative Example 1
Suppose that a function f depends on a variable x which is written as,
$f(x)=\cos x$
We have to calculate the 4th derivative of cos x. The first derivative is,
$\frac{d}{dx}[f(x)]=\frac{d}{dx}[\cos x]$
Since the derivative of cos is -sin,
$f’(x) = -\sin x$
Now to calculate the second derivative, we need to differentiate the first derivative of cos which is –sinx.
$\frac{d}{dx}[f(x)]=\frac{d}{dx}[-\sin x]$
Since the derivative of sin is cos therefore, the second derivative of cos x is,
$f’’(x) = -\cos x$
To calculate third derivative, differentiating second derivative of cos,
$f’’’(x) = \frac{d}{dx}[-\cos x] = \sin x$
Similarly, the fourth derivative of cos is,
$f’’’’(x)=\frac{d}{dx}[\sin x] = \cos x$
Which is the higher order derivative of cos.
Higher Order Derivative example 2
Suppose that we have to calculate the nth derivative of e^x. For this, we will use the following steps.
$y = e^x$
Applying derivative with respect to x,
$\frac{dy}{dx} = \frac{d}{dx}[e^x]$
Since the derivative of an exponential function is equal to the original function multiplied with the derivative of its exponent. So,
$\frac{dy}{dx} = e^x$
Or,
$y_1=e^x$
Where the subscript denotes the number of derivatives of the function. Now again differentiating to get second derivative,
$y_2=e^x$
And the third derivative is,
$y_3=e^x$
Similarly the nth derivative of e is,
$y_n= e^x$
How to find higher order derivatives by using calculator?
The derivative of a function can be also calculated by using a higher order derivative calculator. It is an online tool that follows all derivative formulas to find derivative. You can find it online by searching for a derivative calculator. For example, to calculate the derivative of ln5x, the following steps are used by using this calculator.
- Write the expression of the function in the input box such as, ln5x.
- Choose the variable to calculate the rate of change, which will be x in this example.
- Now choose the value of n. In this step, you need to select the order of derivative.
- Review the input so that there will be no syntax error in the function.
- Now at the last step, click on the calculate button. By using this step, the nth derivative calculator will provide the derivative of ln5x quickly and accurately.
Comparison between Higher order derivative and second derivative test
The comparison between the higher derivative and second derivative test can be easily analysed using the following difference table.
Higher-order Derivative | Second order derivative test |
The process of differentiating a function repeatedly to obtain a specific derivative i.e. nth derivative, is known as higher-order derivative. | The second derivative test is used to determine whether the function is increasing of decreasing. |
The higher order derivative is defined as; $\frac{d^n}{dx^n}[f(x)] = \frac{d}{dx}\left(\frac{d^{n-1}}{dx^{n-1}}f(x)\right)$ | A function is differentiated two time to identify its nature. If f’’(x) >0, the curve will be concave up. Whereas the curve will be concave down if f’’(x)<0. |
The higher order derivative along with different derivative rules can be used to evaluate derivative. | The second derivative test can also be used with derivative rules to find rate of change. |
Conclusion
The higher order derivative is a process of calculating derivative of a function up to n times. We differentiate a function again and again to get the nth derivative. In order word, to calculate the second derivative, the first derivative is differentiated. Therefore, we can conclude that every (n-1)th derivative is differentiated to get the nth derivative.