CalculusMultivariable CalculusCollege

Partial Derivatives

Differentiate with respect to one variable while holding others constant.

A function of several variables, such as $f(x, y) = x^2 y + 3y$ , changes differently depending on which input you vary. The partial derivative $\frac{\partial f}{\partial x}$ measures the rate of change as $x$ varies with $y$ held fixed, and $\frac{\partial f}{\partial y}$ does the reverse. You compute them with the ordinary rules, treating the other variable as a constant.

Together the partial derivatives form the gradient $\nabla f = \left(f_x, f_y\right)$ , which points in the direction of steepest increase. Setting the gradient to zero finds critical points of surfaces, the starting point for optimisation in economics, physics and machine learning.

Key ideas

Hold every other variable constant and differentiate as usual.
Mixed partials $f_{xy}$ and $f_{yx}$ are equal for smooth functions.
The gradient is perpendicular to level curves.
Second derivative test: $D = f_{xx}f_{yy} - f_{xy}^2$ classifies critical points.

Worked example

For $f(x, y) = x^2 y + 3y^2 - 4x$ , find $f_x$ and $f_y$ at $(1, 2)$ .

$f_x = 2xy - 4$ , treating $y$ as constant.
$f_y = x^2 + 6y$ , treating $x$ as constant.
At $(1, 2)$ : $f_x = 4 - 4 = 0$ .
$f_y = 1 + 12 = 13$ .

Answer: $f_x = 0$ , $f_y = 13$

Common mistakes

Differentiating the held-constant variable as if it changed.
Dropping a term like $x^2 y$ entirely when differentiating with respect to $y$ ; it becomes $x^2$ .
Confusing the gradient vector with a single number.

Practice questions

Try each one, then open the answer. Numerical answers on this page are verified by an automated calculation check.

1.For $f = 3x^2 y$ , find $f_x$ at $(1, 4)$ .
Show answer
$24$
2.For $f = xy^3$ , find $f_y$ at $(2, 1)$ .
Show answer
$6$
3.For $f = e^{x} y$ , find $f_x$ at $(0, 5)$ .
Show answer
$5$

Keep practising

Calculators

Partial Derivative Calculator