Partial Differentiation Calculator
Find the partial derivative of any multivariable function instantly. Enter your function and the calculator handles the power rule, product rule, quotient rule, and chain rule automatically.
Function Input
Enter a multivariable function to compute its partial derivative
Use x, y, z as variables. Supported: + − * / ^, sin, cos, tan, exp, log, sqrt. Use * for multiplication, e.g. 3*x*y.
Your Partial Derivative
Derivative, gradient terms, and evaluated value
Enter a function above and click Calculate Partial Derivative to see the result, the gradient, and the evaluated value at your chosen point.
Rules Used in Partial Differentiation
When this calculator finds ∂f/∂x, every other variable in the function is temporarily treated as a constant, and the standard differentiation rules below are applied to the chosen variable only.
Power Rule
∂/∂x (xⁿ) = n·xⁿ⁻¹. Applied to any term where the differentiation variable is raised to a power.
Product Rule
∂/∂x (u·v) = u·∂v/∂x + v·∂u/∂x. Used whenever two expressions involving the variable are multiplied together.
Chain Rule
∂/∂x f(g(x)) = f'(g(x))·∂g/∂x. Applied to nested functions such as sin(xy) or exp(x²).
Quotient Rule
∂/∂x (u/v) = (v·∂u/∂x − u·∂v/∂x) / v². Used for any function written as one expression divided by another.
Common Derivative Rules
A quick lookup of standard derivative forms that appear frequently in partial differentiation problems.
| Function f(x) | Derivative f'(x) | Type | Notes |
|---|---|---|---|
| xⁿ | n·xⁿ⁻¹ | Power Rule | Most common rule in polynomial terms |
| sin(x) | cos(x) | Trigonometric | Multiply by inner derivative for chain rule |
| cos(x) | −sin(x) | Trigonometric | Multiply by inner derivative for chain rule |
| tan(x) | sec²(x) | Trigonometric | Equivalent to 1/cos²(x) |
| eˣ | eˣ | Exponential | Unchanged under differentiation |
| ln(x) | 1/x | Logarithmic | Defined for x > 0 |
| √x | 1/(2√x) | Root | Same as x^(1/2) under the power rule |
| u·v | u·v′ + v·u′ | Product Rule | Used when two functions are multiplied |
| u/v | (v·u′ − u·v′)/v² | Quotient Rule | Used when one function divides another |
Partial Derivative FAQ
Everything you need to know about partial differentiation, mixed partials, and how multivariable derivatives are calculated.
A partial derivative measures how a multivariable function changes as one variable changes while all other variables are held constant. For a function f(x, y), the partial derivative with respect to x, written as ∂f/∂x, treats y as a fixed constant during differentiation.
To calculate a partial derivative with respect to a chosen variable, apply the standard rules of differentiation (power rule, product rule, chain rule, quotient rule) to that variable only, while treating every other variable in the expression as a constant number.
A regular (ordinary) derivative applies to a function of a single variable. A partial derivative applies to a function of two or more variables, and isolates the rate of change with respect to just one of those variables while the rest stay constant.
A mixed partial derivative is a second-order partial derivative taken with respect to two different variables, such as ∂²f/∂x∂y. For most well-behaved functions, Clairaut’s theorem guarantees that the order of differentiation does not matter, so ∂²f/∂x∂y equals ∂²f/∂y∂x.
The gradient of a multivariable function is a vector made up of all its first-order partial derivatives. For f(x, y), the gradient is written as ∇f = (∂f/∂x, ∂f/∂y), and it points in the direction of steepest increase of the function.
Partial derivatives are essential in fields involving multivariable functions, including physics, engineering, economics, and machine learning. They are used to find gradients, optimize functions with several variables, solve partial differential equations, and analyse rates of change in real-world systems.