Use ^ for powers, e.g., x^2. Supported: +, -, *, /, ^, sin, cos, tan, ln, e^x
Power Rule
∂/∂x(x^n) = n·x^(n-1)
Constant Multiple Rule
∂/∂x(c·f) = c·∂f/∂x
Sum Rule
∂/∂x(f+g) = ∂f/∂x + ∂g/∂x
Product Rule
∂/∂x(f·g) = f·∂g/∂x + g·∂f/∂x
Powers: Use ^ for exponents (x^2, y^3)
Products: Use * or implicit (2x, x*y, 3xy)
Functions: sin(x), cos(y), ln(x), e^x
Examples:
- x^2 + y^2
- 3x^2y + 2xy^2
- sin(x) + cos(y)
- e^x + ln(y)
A partial derivative is a derivative of a function of multiple variables with respect to one variable, treating all other variables as constants. It measures the rate of change of the function along one axis while holding other variables fixed. Partial derivatives are fundamental in multivariable calculus and are essential for optimization, physics, engineering, and machine learning applications.
For a function f(x, y), the partial derivative with respect to x, denoted ∂f/∂x, is found by differentiating f treating y as a constant. Similarly, ∂f/∂y is found by treating x as a constant. The collection of all first-order partial derivatives forms the gradient vector ∇f, which points in the direction of steepest ascent of the function.
Optimization
Finding maximum and minimum values of functions by setting partial derivatives to zero (critical points). Used in economics, machine learning, and engineering design.
Physics
Describing physical phenomena like heat flow, wave propagation, and electromagnetic fields through partial differential equations.
Machine Learning
Gradient descent algorithms use partial derivatives to minimize loss functions and train neural networks by computing gradients efficiently.
Economics
Marginal analysis in economics uses partial derivatives to understand how changes in one variable affect outcomes while holding others constant.
Partial derivative calculations follow standard multivariable calculus rules. Results depend on correct function input and selected variable. This calculator handles common algebraic and trigonometric functions. For complex expressions or advanced applications, verify results with additional tools or manual calculation.