Partial Derivative Calculator
Calculate partial derivatives of multivariable functions with step-by-step solutions. Our calculator handles polynomial, trigonometric, and exponential functions while applying differentiation rules and providing detailed mathematical explanations.
Last updated: December 15, 2024
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Partial Derivative Result
Original Function:
Partial Derivative:
First order partial derivative with respect to x
Applicable Rules:
- Power Rule: ∂/∂x(x^n) = n*x^(n-1)
Solution Steps:
- Step 1: Identify the function f(x,y) = x^2 + y^2
- Step 2: Apply power rule to differentiate with respect to x
- Step 3: ∂/∂x(x²) = 2x
- Step 4: Other variables are treated as constants
Quick Example Result
For function f(x,y) = x² + y², the partial derivative with respect to x:
∂f/∂x = 2x
How This Calculator Works
Our partial derivative calculator uses systematic differentiation rules to find partial derivatives of multivariable functions. The process involves treating all variables except the one being differentiated as constants, then applying standard differentiation rules to calculate the rate of change in one direction.
Partial Derivative Notation
First Order:
∂f/∂x, ∂f/∂y, ∂f/∂zSecond Order:
∂²f/∂x², ∂²f/∂x∂y, ∂²f/∂y²Mixed Derivatives:
∂²f/∂x∂y = ∂²f/∂y∂x (when continuous)Shows how the function changes in different variable directions
Mathematical Foundation
Partial derivatives are fundamental to multivariable calculus and are defined as the limit of the difference quotient as one variable approaches zero while others remain fixed. This concept extends single-variable differentiation to functions of multiple variables, enabling analysis of how functions change in multidimensional spaces.
- Power Rule: ∂/∂x(x^n) = n·x^(n-1) (treating other variables as constants)
- Product Rule: ∂/∂x(uv) = u(∂v/∂x) + v(∂u/∂x) where u,v are functions of x
- Chain Rule: ∂/∂x(f(g(x,y))) = f'(g)·(∂g/∂x)
- Constant Rule: ∂/∂x(c) = 0 where c doesn't depend on x
Sources & References
- Stewart Multivariable Calculus - Partial Derivatives ChapterComprehensive coverage of multivariable differentiation
- MIT OpenCourseWare - Multivariable Calculus Course MaterialsDetailed examples and applications of partial derivatives
- Wolfram MathWorld - Partial Derivative ReferenceMathematical encyclopedia with examples and proofs
Need help with other calculus calculations? Check out our derivative calculator and area between curves calculator.
Get Custom Calculator for Your BusinessExample Calculation
Given Function:
- Function: f(x,y) = x² + y²
- Variable: x
- Order: First derivative
Solution Steps:
- Identify: f(x,y) = x² + y²
- Differentiate x² with respect to x: 2x
- Treat y² as constant: derivative is 0
- Result: ∂f/∂x = 2x
Result: ∂f/∂x = 2x
This shows how the function changes with respect to x while keeping y constant.
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