Partial Derivative Calculator
Compute partial derivatives of multivariable functions with step-by-step explanations
Compute Partial Derivatives
Enter your function and variable to differentiate with respect to
Result
Function: x^2 + y^2
Differentiate with respect to: x
Order: First Derivative
About Partial Derivatives
What is a Partial Derivative?
A partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant. It measures how the function changes as only that variable changes.
How to Use This Calculator
1. Enter your function using standard mathematical operations and functions (e.g., sin, cos, exp, log).
2. Specify the variable you want to differentiate with respect to (e.g., x, y, z).
3. Select the order of derivative (1st, 2nd, or 3rd).
4. Click “Calculate Partial Derivative” to get the result.
Examples
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f(x,y) = x² + y²∂f/∂x = 2x, ∂f/∂y = 2y
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f(x,y) = x * y + sin(x)∂f/∂x = y + cos(x), ∂f/∂y = x
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f(x,y) = e^(x*y)∂f/∂x = y * e^(x*y), ∂f/∂y = x * e^(x*y)
Partial Derivative Calculator – Multivariable Calculus Tool
Our partial derivative calculator is an essential tool for students, educators, and professionals working with multivariable calculus. It simplifies the process of finding partial derivatives of complex functions with respect to specific variables, saving time and reducing calculation errors.
Partial derivatives are fundamental in vector calculus and have applications in physics, engineering, economics, and machine learning. They are used to analyze surfaces, optimize functions with multiple variables, and solve partial differential equations.
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How Partial Derivatives are Calculated
The calculator uses symbolic differentiation to compute partial derivatives. When you input a function f(x,y,z,…) and specify a variable (e.g., x), the calculator:
1. Parses the mathematical expression
2. Applies differentiation rules (power rule, product rule, quotient rule, chain rule)
3. Treats all other variables as constants
4. Simplifies the resulting expression
Applications of Partial Derivatives
– Gradient Vector: Partial derivatives form the components of the gradient vector (∇f), which points in the direction of the steepest ascent of the function.
– Optimization: Used in finding maximum and minimum values of multivariable functions (critical points).
– Physics: Appear in thermodynamics, electromagnetism, and fluid dynamics equations.
– Economics: Marginal costs and production functions often involve partial derivatives.
– Machine Learning: Essential for gradient descent algorithms in training neural networks.