nth Derivative calculator

Curious about how to calculate the nth derivative of a function quickly? Discover how our nth derivative calculator can make this complex task easier for you!

with respect to

Feedback

How easy was it to use our calculator? Did you face any problem tell us!

Get the Widget

Add this powerfull tool to your website and help them balance chemical equations in second

Introduction to nth Derivative Calculator:

The nth derivative calculator is an amazing tool used to compute the nth derivative of a given function concerning its independent variable. It simplifies the process of calculating higher derivatives by providing insights into the nature of functions.

nth Derivative Calculator with Steps

The higher order derivative calculator is a valuable tool for anyone who needs to compute higher order derivatives quickly and efficiently. It is used in signal processing, control theory, and mechanics to analyze rates of change.

What is nth Derivative?

The nth derivative of a function f(x) involves successively finding higher order derivatives n times. It is denoted as fⁿ(x). This method goes beyond the first and second order, and it is used in calculus to analyze higher-order changes in functions.

The nth derivative method is used in numerical methods and algorithms for solving differential equations and optimization problems because it has a wide range of applications in various scientific and engineering disciplines. Mathematically the definition of nth derivative from the given function f(x) is,

$$ f^0 (x) \;=\; f(x)\; (the\; function\; itself) $$

$$ f^1 (x) \;=\; \frac{d}{dx} f(x)\; (the\; first\; derivative) $$

$$ f^2 (x) \;=\; \frac{d}{dx} f^1 (x) \;=\; \frac{d^2}{dx^2} f(x)\; (the\; second\; derivative) $$

$$ \vdots $$

$$ f^n (x) \;=\; \frac{d}{dx} f^{n-1}(x) \;=\; \frac{d^n}{dx^n} f(x)\; (the\; nth\; derivative) $$

How to Find the nth Derivative?

For finding the nth derivative of a function, you can use our nth derivative calculator for a quick automated solution in seconds. Alternatively, you can manually apply the derivative operator from first order to nth times successively. Here is the method to find the nth derivative of a function f(x).

Step-by-Step Method:

Step 1: Start with the original function: Let f(x) be the function whose nth derivative fⁿ(x) you want to find.

Step 2: Take the first derivative and apply the differentiation rules like power rule, product rule, chain rule, etc.

Step 3: Take the second derivative and again, apply the differentiation rules to find f²(x).

Step 4: Again take the Third Derivative: f³(x) of the given function with respect to its differential variable.

Step 5: This process will continue till the nth derivative: fⁿ(x) and get the solution differential function.

Example of Higher Order Derivative Calculator:

The practical example of the nth derivative function will give you a better understanding about the working process of higher order derivatives calculator.

Example: Let’s Find the nth Derivative of Following:

$$ f(x) \;=\; x^n $$

n is the positive integer.

Solution:

For the solution of nth derivative, you need to differentiate from first order derivative to nth derivative function. Differentiation the function f(x) using power rule with respect to x,

$$ f’(x) \;=\; \frac{d}{dx}x^n \;=\; nx^{x-1} $$

Again differentiation f`(x),

$$ f’’(x) \;=\; \frac{d}{dx} (nx^{n-1}) \;=\; n(n-1)x^{n-2} $$

Differentiate the second order derivative function,

$$ f’’’(x) \;=\; \frac{d}{dx} (n(n-1) x^{n-2}) \;=\; n(n-1)(n-2)x^{n-3} $$

The derivation will continuous till the nth derivative not achieved,

$$ f^n(x) \;=\; n(n-1)(n-2) … (n-(n-1)) x^{n-n} \;=\; n! $$

So, the nth derivative of f(x)= xⁿ f(x) is fⁿ(x)=n!, where n! denotes the factorial of n.

How to Use the Higher Derivative Calculator?

Multiple derivative calculator has a user-friendly interface that makes it easy for you to know how to use it for the evaluation of nth derivative problem. Follow some instructions that are given as:

Choose the variable of nth derivation of the given field to find the singular value.
Enter the nth derivative function to find the nth derivative solution in the input field.
Review the given problem before hitting the calculate button to start the evaluation process in the higher derivatives calculator.
Click the “Calculate” button to get the result of your given nth derivative problem.
If you are trying our nth order derivative calculator for the first time then you can use the load example to learn more about this method.
Click on the “Recalculate” button to get a new page for finding more example solutions of nth derivative problems.

Final Result of nth Derivative Calculator:

The higher order derivative calculator give you the solution from a given nth derivative problem when you add the input into it that may contain as:

Result Option:

When you click on the result option it gives you a solution to the nth derivative problem.

Possible Steps:

When you click on it, this option will provide you with a solution where all the calculations of the nth derivative process are included.

Advantages of Using Higher Order Derivatives Calculator:

Higher derivative calculator provides you with many advantages that help you to calculate the nth derivative problems and give you solutions. These advantages are:

It is a free tool so you can use it for free to find the nth derivative problem with solutions without charging.
Multiple derivative calculator is a manageable tool that can manage various types of nth derivative problem to calculate the solution.
Our tool helps you to get conceptual clarity for the nth derivative process when you use it for solving different examples.
Higher derivatives calculator saves the time that you consume on the calculation of the nth derivative problems.
It is a trustworthy tool that provides you with accurate solutions whenever you use it to calculate the nth derivative problem without any error.
The nth order derivative calculator provides the solution without imposing any condition of signup after two to three usage for the evaluation.

Related References

How would you define nth derivative?
Explain the evaluation process of nth derivative:
Give some examples of nth derivative:
Give an overview of nth derivative?

Frequently Ask Questions

How to find the nth derivative of cosx

For Finding the nth derivative of cos(x) follows some simple steps of cosine functions. Here’s a step-by-step method to find the nth derivative of cos(x).

Solution:

Identify the given function f(x),

$$ f(x) \;=\; cos(x) $$

Differentiate the function f(x) with respect to x,

$$ f’(x) \;=\; -sin(x) $$

The process of differentiation is continuous till the nth derivative solution is achieved. The second derivative is,

$$ f’’(x) \;=\; -cos(x) $$

The third derivative is,

$$ f’’’(x) \;=\; sin(x) $$

Fourth derivative is,

$$ f^4(x) \;=\; cos(x) $$

The nth derivative of the given function is f^n(x),

$$ \frac{\partial^n cos(x)}{\partial x^n} \;=\; cos(\frac{n π}{2} + x)\; for\; (n ∈ Z \;and\; n ≥ 0) $$

How to find nth derivative of e^x sinx

$$ f(x) \;=\; e^x sin(x) $$

Differentiate the given function with respect to x successively till the nth derivative is not achieved

Apply the product rule: (uv)’ = u’v + uv’:

$$f’(x) \;=\; (e^x sin(x))’ \;=\; e^x sin(x) + e^x cos(x) $$

Second derivative:

Differentiate f’(x) again using the product rule:

$$ f’’(x) \;=\; (e^x sin(x) + e^x cos(x))’ \;=\; e^x sin(x) + 2e^x cos(x) - e^x sin(x) \;=\; 2e^x cos(x) $$

Third derivative:

$$ f’’’(x) \;=\; (2e^x cos(x))’ \;=\; 2(e^x cos(x))’ \;=\; 2(e^x cos(x) - e^x sin(x)) \;=\; 2e^x (cos(x) - sin(x)) $$

Fourth derivative:

$$ f^4 (x) \;=\; (2e^x (cos (x) - sin(x)))’ \;=\; 2(e^x (cos(x) - sin(x)))’ \;=\; 2(e^x (cos(x) - sin(x)) + e^x (-sin(x) - cos(x))) $$

$$ =\; 2(e^x cos(x) - e^x sin(x) - e^x sin(x) - e^x cos(x)) \;=\; -4e^x sin(x) $$

For the nth derivative, the process will continue. The general solution of differential function is,

$$ f^n (x) \;=\; -2^{\frac{n}{2}} e^x cos(x) $$

Where n = 0,1,2,3,...

How to find the nth derivative of a power series

To find the nth derivative of a power series, let's start with the general form of a power series. Then apply the differentiation. Here’s a step-by-step

A general form of power series centered at x = a is,

$$ f(x) \;=\; \sum_{k=0}^∞ c_k (x - a)^k $$

Differentiate the series successively till nth order:

$$ f’(x) \;=\; \sum_{k=1}^∞ c_k . k . (x - a)^{k-1} $$

Second derivative,

$$ f’’(x) \;=\; \sum_{k=2}^∞ c_k . k . (k - 1) . (x - a)^{k-2} $$

For the nth derivation,

$$ f^n (x) \;=\; \sum_{k=n}^∞ c_k . k . (k - 1) . (k - 2) … (k - n + 1) . (x - a)^{k-n} $$

What is the nth derivative of logx

To find the nth derivative of log⁡(x), differentiating log⁡(x) repeatedly from first order to nth order. Let's go through the process step-by-step to find the general formula.

$$ f(x) \;=\; log(x) $$

For first derivative,

$$ f’(x) \;=\; \frac{1}{x} $$

Second derivative,

$$ f’’(x) \;=\; \frac{d}{dx} (\frac{1}{x}) \;=\; - \frac{1}{x^2} $$

Third derivative,

$$ f’’’(x) \;=\; \frac{d}{dx} (- \frac{1}{x^2}) \;=\; 2 . \frac{1}{x^3} \;=\; \frac{2}{x^3} $$

Fourth derivative,

$$ f^4 (x) \;=\; \frac{d}{dx} (\frac{2}{x^3}) \;=\; -3 . \frac{2}{x^4} \;=\; - \frac{6}{x^4} $$

For the nth derivative, the general form becomes,

$$ f^n (x) \;=\; (-1)^{n-1} \frac{(n-1)!}{x^n} $$

Where n varies n=0,1,2,3,..

How to find the nth derivative of e^ x

The function e^x has a unique property: its derivative is the same as the original function. This property simplifies the process of finding higher-order derivatives of e^x.

For first derivative

$$ f’(x) \;=\; e^x $$

Assume a supposition you want to find the kth derivative then the function is,

$$ f^k (x) \;=\; e^x $$

If you take k+1 then its derivative remains the same,

$$ f^(k+1) (x) \;=\; \frac{d}{dx} e^x \;=\; e^x $$

For the nth derivative the general form becomes,

$$ f^n (x) \;=\; e^x $$

Amanda Jepson

Last Updated February 21, 2024