nth Derivative Calculator

Category: Calculus

- June 21, 2025

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This calculator finds the nth derivative of a function. Enter your function, specify the order of the derivative, and see the step-by-step differentiation process.

Input Function

Function f(x):

Order of Derivative (n):

Differentiation Variable:

Evaluate at point (optional):

Display Options

Decimal Places:

Show step-by-step solution

Show graph

Derivative Results

nth Derivative f'(x)

f^(n)(x)

d^n f(x)/dx^n

Derivative Sequence

Order	Notation	Derivative	Value at Point

Differentiation Steps

Function Visualisation

Original Function f(x)

nth Derivative f^(n)(x)

Mathematical Notation

About Higher-Order Derivatives

The nth derivative represents the result of differentiating a function n times. Higher-order derivatives are important in calculus, physics, engineering, and other scientific fields for analysing rates of change, approximating functions, and solving differential equations.

Notation for Higher-Order Derivatives

Leibniz Notation: dⁿf/dxⁿ or dⁿy/dxⁿ
Lagrange Notation: f⁽ⁿ⁾(x)
Newton Notation: y⁽ⁿ⁾ or f⁽ⁿ⁾

Common Examples

Function	First Derivative	Second Derivative	nth Derivative
x^m	mx^m-1	m(m-1)x^m-2	m!/(m-n)! · x^m-n for n ≤ m
e^ax	ae^ax	a²e^ax	aⁿe^ax
sin(ax)	a·cos(ax)	-a²·sin(ax)	aⁿ·sin(ax + nπ/2)
cos(ax)	-a·sin(ax)	-a²·cos(ax)	aⁿ·cos(ax + nπ/2)

Applications of Higher-Order Derivatives

Taylor Series: Approximating functions using higher-order derivatives at a point
Analysing Motion: Position → Velocity (1st derivative) → Acceleration (2nd derivative) → Jerk (3rd derivative) → Snap (4th derivative)
Curve Analysis: Studying concavity, inflection points, and local behaviour
Differential Equations: Solving higher-order differential equations in physics and engineering

Understanding the nth Derivative Calculator

The nth Derivative Calculator is a helpful tool for anyone studying Calculus. It simplifies the process of finding the nth derivative of a function. Just type in your function, choose how many times you want to differentiate it, and let the calculator do the heavy lifting. This makes it easier for students and professionals to understand complex calculus concepts.

How to Use the Calculator

This calculator is straightforward. You start by entering a function, like x² or sin(x). Then, you specify the order of the derivative you want, such as 1 for the first derivative or 3 for the third derivative. If you want, you can also evaluate the derivative at a specific point, which can be useful for applications in Physics and engineering.

Step-by-Step Solutions

One of the standout features of this calculator is its ability to show the steps involved in finding the derivative. This is great for learners who want to understand how differentiation works. You can see how the derivative is calculated, making it easier to follow along and learn the process.

Visualise with Graphs

Another useful feature is the ability to see graphs of both the original function and its nth derivative. This visual representation helps in understanding how functions change as you take higher-order derivatives. It’s beneficial for grasping concepts like concavity and points of inflection.

Choosing Display Options

The nth Derivative Calculator allows you to customise your experience. You can choose how many decimal places you want in your results. Plus, you can decide if you want to see the step-by-step solution and the graph. This customisation makes it a versatile tool tailored to your needs.

Notation of Higher-Order Derivatives

Leibniz Notation: This is written as dⁿf/dxⁿ.
Lagrange Notation: This uses f⁽ⁿ⁾(x).
Newton Notation: Often shown as y⁽ⁿ⁾.

Understanding these notations is key in calculus. They provide different ways to express the same mathematical idea, and knowing them can help you read and write mathematical expressions more easily.

Importance of Higher-Order Derivatives

Taylor Series: Used for approximating functions.
Motion Analysis: Helps in understanding velocity and acceleration.
curve analysis: Important for studying graphs and their shapes.
Differential Equations: Crucial in physics and engineering for modelling real-world problems.

Higher-order derivatives play a significant role in various fields. They help solve complex problems and provide deeper insights into the behaviour of functions.

Conclusion and Applications

The nth Derivative Calculator is an invaluable resource for students, teachers, and professionals. It helps clarify the complexities of calculus, making higher-order derivatives easier to grasp. Whether you’re working on homework or tackling real-world problems, this calculator can guide you through the process, ensuring you get the most accurate and understandable results.