nth Derivative Calculator

Category: Calculus

- April 29, 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 Visualization

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 analyzing 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
Analyzing Motion: Position → Velocity (1st derivative) → Acceleration (2nd derivative) → Jerk (3rd derivative) → Snap (4th derivative)
Curve Analysis: Studying concavity, inflection points, and local behavior
Differential Equations: Solving higher-order differential equations in physics and engineering

General Formula for the nth Derivative:

\\( \frac{d^n}{dx^n} f(x) \\)

What is the nth Derivative Calculator?

The nth Derivative Calculator is a simple and effective tool to find higher-order derivatives of a function. Whether you want the first, second, or any nth derivative, this calculator provides instant results. It supports differentiation for various functions like polynomials, exponentials, and trigonometric functions. The tool also offers step-by-step solutions and visual graphs to help users understand the differentiation process clearly.

Key Features

Compute higher-order derivatives quickly and easily
Optional evaluation at specific points
Displays clear step-by-step differentiation
Graphical visualization of original and differentiated functions
Adjust decimal precision for results
Supports common notations like Leibniz and Lagrange

How to Use the nth Derivative Calculator

Step 1: Enter your function (for example, x^3, sin(x), or e^x).
Step 2: Specify the order of the derivative (e.g., 1 for first, 2 for second).
Step 3: Confirm the variable you are differentiating with respect to (default is x).
Step 4: (Optional) Provide a value to evaluate the derivative at a specific point.
Step 5: Choose if you want to view step-by-step solutions and a graph.
Step 6: Click Calculate Derivative to get your results instantly.

Benefits of Using This Calculator

Using the nth Derivative Calculator can help:

Enhance understanding of higher-order derivatives and their patterns
Prepare for topics like Taylor Series and Curve Analysis
Support homework, research, and real-world problem-solving
Quickly verify manual differentiation work
Visualize complex functions and their derivatives for better intuition

Related Calculators You Might Find Useful

Partial Derivative Calculator for multivariable differentiation and to compute partials
Second Derivative Calculator to find second derivatives and perform advanced derivative analysis
Directional Derivative Calculator to compute directional derivatives in a vector field
Antiderivative Calculator to find antiderivatives and solve integration problems
Derivative Calculator for fast instant derivative results and solve derivatives online

FAQ

What is a higher-order derivative?

A higher-order derivative is the result of differentiating a function multiple times. The first derivative shows the rate of change, the second shows the rate of change of the rate of change, and so on.

Can I use the calculator for partial derivatives?

No, this calculator is for single-variable functions. To compute partial derivatives, use a Partial Derivative Calculator or a multivariable derivative solver.

What if I need the second derivative?

You can simply set the derivative order to 2. Alternatively, the Second Derivative Calculator can find second derivatives and perform second order derivative analysis more directly.

Can I visualize the function and its derivative?

Yes. The calculator generates a clear graph of the original function and its nth derivative to help you better understand how the function behaves.

Is there a tool for finding antiderivatives?

Yes, if you need to find antiderivatives or solve integrals online, you should use an Antiderivative Calculator or an Integral Solver.

How This Calculator Can Help You

The nth Derivative Calculator can be an essential part of your study or work toolkit. Whether you are solving problems for class, analyzing motion in physics, or studying function behavior, having instant access to compute nth derivatives saves time and builds deeper understanding. It complements tools like the Partial Derivative Calculator for multivariable functions, the Second Derivative Calculator for curve analysis, and the Antiderivative Calculator for integration tasks.