Partial Derivative Calculator
Step-by-Step Solver
Compute partial derivatives instantly for any multivariable function with step-by-step solutions and 3D visualization.Used by 500K+ students • Supports all mathematical functions
Partial Derivative Calculator
Compute partial derivatives with step-by-step solutions
Function Input
Ready to Calculate Partial Derivatives?
Enter your multivariable function and select the variable to get instant results
Understanding Partial Derivatives
Learn the fundamentals of multivariable calculus and partial differentiation
What are Partial Derivatives?
Partial derivatives measure how a multivariable function changes with respect to one variable while keeping all other variables constant. They're essential for understanding rates of change in multiple dimensions.
Geometric Interpretation
Geometrically, ∂f/∂x represents the slope of the surface z = f(x,y) in the x-direction. It's like taking a cross-section of the 3D surface and finding the slope of the resulting curve.
Calculation Rules
When calculating ∂f/∂x, treat all other variables (y, z, etc.) as constants. Apply the same differentiation rules as single-variable calculus: power rule, product rule, chain rule, etc.
How the Partial Derivative Calculator Works
Our calculator uses symbolic differentiation to compute exact partial derivatives
Example: Find ∂f/∂x for f(x, y) = 3x²y + sin(xy)
Identify the function and variable
f(x, y) = 3x²y + sin(xy), finding ∂f/∂x
Treat y as a constant
Since we're differentiating with respect to x, y is treated as a constant
Apply differentiation rules
∂/∂x(3x²y) = 6xy (power rule), ∂/∂x(sin(xy)) = y·cos(xy) (chain rule)
Combine results
∂f/∂x = 6xy + y·cos(xy)
Symbolic vs Numerical Methods
Our calculator uses symbolic differentiation, which means it manipulates mathematical expressions algebraically to find exact derivatives, not approximations.
- Exact results, not approximations
- Works with any mathematical function
- Shows step-by-step process
Supported Functions
Our calculator supports a wide range of mathematical functions commonly used in calculus and engineering.
Basic Functions
- • Polynomials (x^n)
- • Exponentials (e^x)
- • Logarithms (ln, log)
- • Square roots (sqrt)
Trigonometric
- • sin, cos, tan
- • sec, csc, cot
- • Inverse functions
- • Hyperbolic functions
Applications of Partial Derivatives
Discover how partial derivatives are used across various fields and industries
Machine Learning
Gradient descent algorithms use partial derivatives to minimize cost functions and train neural networks.
Economics
Marginal analysis uses partial derivatives to study how changes in one variable affect economic outcomes.
Physics
Heat transfer, wave equations, and electromagnetic field calculations rely heavily on partial derivatives.
Engineering
Stress analysis, fluid dynamics, and optimization problems in engineering use partial derivatives extensively.
Real-World Example: Optimization
Finding Maximum Profit
A company's profit function P(x,y) depends on two products. To maximize profit:
- 1.Find ∂P/∂x = 0 and ∂P/∂y = 0
- 2.Solve the system of equations
- 3.Use second derivatives to confirm maximum
3D visualization of profit optimization
Types of Partial Derivatives Supported
Our calculator handles various types of partial derivatives and multivariable functions
First-Order Partial Derivatives
Basic partial derivatives like ∂f/∂x, ∂f/∂y, and ∂f/∂z for functions of multiple variables.
∂f/∂x, ∂f/∂y, ∂f/∂zHigher-Order Derivatives
Second and higher-order partial derivatives for advanced analysis and optimization problems.
∂²f/∂x², ∂²f/∂y²Mixed Partial Derivatives
Mixed derivatives involving multiple variables, useful for studying interaction effects.
∂²f/∂x∂y, ∂²f/∂y∂xPolynomial Functions
Multivariable polynomials of any degree with multiple variables and cross terms.
x³y² + 2xy + zTrigonometric Functions
Sine, cosine, tangent and their combinations with multiple variables.
sin(xy), cos(x+y)Exponential & Logarithmic
Exponential and logarithmic functions with multiple variables and compositions.
e^(xy), ln(x+y)User Guide: How to Use the Calculator
Step-by-step instructions for getting the most out of our partial derivative calculator
Enter Your Function
Type your multivariable function using standard mathematical notation. Use ^ for powers, * for multiplication.
Examples: x^2*y, sin(x*y), e^(x+y), ln(x^2+y^2)
Select Variable and Order
Choose which variable to differentiate with respect to (x, y, or z) and select first or second order.
Tip: Start with first-order derivatives before moving to higher orders
Review Results
Examine the simplified result and step-by-step solution. Use the comparison feature to analyze multiple derivatives.
Features: PDF export, sharing, and detailed explanations
Function Input Syntax
Basic Operations
x + yx - yx*y or xyx/yx^2sqrt(x)Advanced Functions
sin(x)cos(x)ln(x)e^xCommon Examples
- •
x^2 + y^2- Simple polynomial - •
sin(x*y)- Trigonometric composition - •
e^(x+y)- Exponential function - •
ln(x^2+y^2)- Logarithmic composition
Recommended Study Materials
Essential textbooks and tools to master partial derivatives and multivariable calculus
Multivariable Calculus Textbooks
- Comprehensive coverage of partial derivatives
- Step-by-step examples and exercises
Scientific Graphing Calculator
- 3D plotting and visualization
- Symbolic computation capabilities
Mathematical Software Guides
- Learn Mathematica, MATLAB, or Maple
- Advanced computational techniques
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Coming SoonLearn More About Partial Derivatives
Expand your knowledge with these authoritative educational resources
Wolfram MathWorld
Comprehensive mathematical reference
Explore the complete mathematical theory behind partial derivatives, including formal definitions, theorems, and advanced applications in mathematics and physics.
Visit MathWorldKhan Academy
Free online learning platform
Learn multivariable calculus from the ground up with interactive exercises, video lessons, and practice problems covering partial derivatives and more.
Start LearningFrequently Asked Questions
Common questions about partial derivatives and our calculator
Master Partial Derivatives Today
Use our calculator to solve complex multivariable functions and understand the step-by-step process