Our Partial Derivative Calculator helps you instantly compute partial derivatives of multivariable functions with clear, step-by-step explanations. Whether you’re a student learning calculus or a professional checking complex equations, this free tool simplifies differentiation with respect to any variable in seconds.
What is a Partial Derivative?
A partial derivative represents the rate of change of a multivariable function with respect to one variable while keeping all other variables constant. Unlike ordinary derivatives used in single-variable calculus, partial derivatives are essential for analyzing functions that depend on multiple independent variables.
For example, if you have a function f(x,y) that represents the temperature at different points on a surface, the partial derivative ∂f/∂x tells you how quickly the temperature changes as you move in the x-direction, while keeping y constant. This is denoted using the symbol ∂ (partial) instead of d to distinguish it from ordinary derivatives.
Partial derivatives form the foundation of multivariable calculus and are crucial in optimization problems, gradient descent algorithms in machine learning, physics equations, and economic modeling.
How to Use This Partial Derivative Calculator
Our calculator makes finding partial derivatives simple and educational. Follow these steps:
- Enter your function: Type your multivariable function using standard mathematical notation. For example: x^2*y^3, sin(x*y), e^x + e^y, or x^3*y^2*z
- Specify the derivative order: Enter the variables you want to differentiate with respect to, in order. Examples:
- “x” – first derivative with respect to x
- “x,y” – mixed partial derivative, first x then y
- “x^2” – second derivative with respect to x
- “x^2,y” – second derivative w.r.t. x, then first derivative w.r.t. y
- Click Calculate: Get instant results with detailed step-by-step explanations
- Explore micro-steps: Expand each step to see granular details of the calculation
- Verify numerically: Test your derivative at specific points to ensure accuracy
Supported Functions
Our calculator supports a wide range of mathematical functions including:
- Polynomials: x^2, x^3*y^2, 5*x*y*z
- Exponential: e^x, exp(x*y), e^(x^2)
- Trigonometric: sin(x), cos(x*y), tan(x)
- Logarithmic: log(x), ln(x*y)
- Mixed expressions: x^2*sin(y) + e^x*cos(y)
Essential Partial Derivative Rules
Understanding these fundamental rules will help you master partial derivatives:
Power Rule
∂/∂x(x^n) = n·x^(n-1)
Bring down the exponent and reduce it by 1
Constant Multiple Rule
∂/∂x(c·f) = c·∂f/∂x
Constants can be factored out
Sum/Difference Rule
∂/∂x(f + g) = ∂f/∂x + ∂g/∂x
Derivative of sum = sum of derivatives
Product Rule
∂/∂x(f·g) = (∂f/∂x)·g + f·(∂g/∂x)
First times derivative of second + second times derivative of first
Chain Rule
∂/∂x(f(g(x))) = f'(g)·g'(x)
Derivative of outer times derivative of inner
Mixed Partials
∂²f/∂y∂x = ∂²f/∂x∂y
Order doesn’t matter (if continuous)
Real-world Use Cases
Physics & Engineering
- Heat transfer and thermodynamics equations
- Fluid dynamics and wave propagation
- Electromagnetic field calculations
- Structural optimization and stress analysis
Economics & Finance
- Marginal utility in consumer theory
- Production function optimization
- Risk analysis in portfolio management
- Price elasticity calculations
Machine Learning & AI
- Gradient descent optimization algorithms
- Backpropagation in neural networks
- Loss function minimization
- Feature sensitivity analysis
Statistics & Data Science
- Maximum likelihood estimation
- Regression analysis and curve fitting
- Multivariate probability distributions
- Sensitivity analysis in models
Related Mathematical Concepts
Understanding partial derivatives opens the door to many advanced mathematical concepts:
- Gradient Vector: The vector of all first-order partial derivatives, indicating the direction of steepest ascent
- Directional Derivative: Rate of change in any specified direction, not just along coordinate axes
- Jacobian Matrix: Matrix of all first-order partial derivatives for vector-valued functions
- Hessian Matrix: Matrix of second-order partial derivatives, used in optimization and curvature analysis
- Total Derivative: Combines partial derivatives when all variables are functions of another variable
- Implicit Differentiation: Finding derivatives when variables are related implicitly rather than explicitly
About This Partial Derivative Calculator
Our free partial derivative calculator provides instant, accurate results with detailed step-by-step explanations. Whether you’re a student learning multivariable calculus, a professional working on optimization problems, or a researcher analyzing complex functions, our tool breaks down each calculation into easy-to-understand micro-steps. Supporting all common mathematical functions including polynomials, trigonometric, exponential, and logarithmic expressions, this calculator helps you master partial derivatives through clear explanations and alternate solution methods.
What is a partial derivative?
A partial derivative measures how a multivariable function changes with respect to one variable while keeping the others constant. It’s commonly used in calculus, optimization, and machine learning to analyze how each variable affects the overall function.
How do I use the Partial Derivative Calculator?
Simply enter a multivariable function such as x²y + 3xy², select the variable you want to differentiate (e.g., x or y), and click Calculate Partial Derivative. The tool will instantly display the result along with detailed, step-by-step explanations.
Can this tool handle more than two variables?
Yes. The calculator supports functions with multiple variables (e.g., f(x, y, z)), so you can easily find partial derivatives like ∂f/∂x, ∂f/∂y, or ∂f/∂z.
Does the calculator show step-by-step solutions?
Absolutely. Each computation includes a clear, step-by-step breakdown showing how the derivative was computed and simplified — perfect for learning or verifying manual work.
Is this Partial Derivative Calculator free to use?
Yes, it’s completely free — no sign-up, ads, or limits. You can calculate as many derivatives as you want directly from your browser.