Partial Derivative Calculator (Multivariable)

What a partial derivative measures

A partial derivative answers one question: if you nudge a single input and freeze everything else, how fast does the output move? For a surface z = f(x, y), the partial f_x is the slope you would feel walking due east across that surface, and f_y is the slope walking due north. Neither on its own is the steepest slope — that is the job of the gradient vector, which packages the partials into a direction.

The freezing is the whole idea. When you differentiate x^2*y with respect to x, the y is a constant multiplier and the answer is 2xy. Differentiate the same expression with respect to y and now x² is the constant multiplier, giving x². One expression, two completely different derivatives, because two different things were held still.

That is why partials show up wherever a quantity depends on several things at once. A thermodynamicist writes (∂U/∂V)_S and puts the held-constant variable in a subscript, because the same derivative taken at constant temperature is a different number. An economist reads ∂Q/∂p as the response of demand to price with income unchanged. A machine-learning gradient step is nothing but a vector of partials of the loss with respect to each weight.

The rules are the single-variable rules, applied one variable at a time

You already know every rule you need. Treat the other variables as if they were the number 7, then differentiate normally.

Power rule. ∂/∂x of x³y² is 3x²y². The y² rides along untouched.

Product rule. It only applies when both factors contain the variable you are differentiating. In x²·sin(xz), differentiating in x needs the product rule; differentiating in y gives zero because neither factor contains y.

Chain rule. For sin(yz) differentiated in y, the outer derivative is cos(yz) and the inner derivative of yz in y is z, so the answer is z cos(yz). Forgetting that inner factor is the single most common error in a Calculus III exam.

Second and mixed partials. Differentiate twice. f_xx means differentiate in x and then in x again; f_xy means differentiate in x first and then in y. The notation is genuinely inconsistent across textbooks: the subscript form f_xy reads left to right, while the operator form ∂²f/∂y∂x reads right to left. They mean the same thing. This page uses the subscript order, so f_xy is x first.

Clairaut's theorem (also attributed to Schwarz and to Young in different texts) says the order does not matter as long as the second partials are continuous near the point. That is a genuine theorem with a genuine hypothesis, and the calculator reports the largest disagreement it finds so you can see the symmetry rather than assume it.

How to read the numbers you get back

Sign is the first thing to read. A positive f_x means f increases as x increases from this point; a negative one means it decreases. The size tells you the rate per unit of x, in whatever units f carries.

All first partials zero marks a critical point. That is where maxima, minima and saddles live. Which one you have is decided by the second partials through the discriminant D = f_xxf_yy − f_xy²: if D > 0 and f_xx > 0 you have a local minimum, if D > 0 and f_xx < 0 a local maximum, and if D < 0 a saddle. When D = 0 the test is silent and tells you nothing either way. At the default point the partials are not zero, so no classification applies there.

Units matter more than students expect. If f is pressure in kilopascals and x is temperature in kelvin, then f_x is in kPa per K. A second partial f_xx is in kPa per K squared. Reporting a partial without its units is reporting half an answer.

The mixed partial has a plain meaning: it is how much the x-slope changes when you move in y. A zero mixed partial means the two variables act independently on the slope, which is exactly the case for any function that splits as g(x) + h(y).

Common functions and their partials

The f_xy column differentiates in x first and then in y. Every entry is symmetric in the sense that differentiating in the other order gives the same expression.

Mistakes that cost marks

• Dropping the inner derivative. ∂/∂y of sin(yz) is z cos(yz), not cos(yz). The chain rule does not switch off just because you are holding another variable constant.
• Differentiating the held variable anyway. In x²y³ differentiated in x, the y³ is a constant. It does not become 3y².
• Reading the operator notation backwards. ∂²f/∂y∂x is x first, then y — the operator nearest f acts first. The subscript form f_xy is the opposite reading order for the same object.
• Assuming Clairaut always applies. It needs continuity of the second partials near the point. The textbook counterexample xy(x²−y²)/(x²+y²) has f_xy = −1 and f_yx = +1 at the origin, and that function is not one this calculator can be given because the expression is undefined there.
• Working in degrees. Every trigonometric function here takes radians, which is the only convention under which d(sin u)/du = cos u holds.
• Evaluating at a point outside the domain. ln(x) at x = −2 or 1/z at z = 0 has no value, and neither does its derivative; the calculator reports that rather than inventing a number.

Where partial derivatives lead next

The partials are the raw material for almost everything in multivariable calculus. Stack them into a vector and you have the gradient, whose direction is the steepest ascent and whose length is the steepest slope; the directional derivative in any unit direction u is just the dot product ∇f · u. Stack the partials of a vector field into a matrix and you have the Jacobian, whose determinant is the local volume-scaling factor used in every change-of-variables integral — see the Jacobian determinant calculator. Take the partials of a vector field and combine them one way and you get the divergence; combine them another way and you get the curl.

Second partials assemble into the Hessian matrix, whose eigenvalues classify critical points in any number of variables — the two-variable discriminant test above is just the two-by-two determinant of that matrix. If you want those eigenvalues, the eigenvalue calculator takes the Hessian directly.

If your function has only one variable, none of this machinery is needed: use the ordinary derivative calculator or the second derivative calculator. And if what you actually need is to integrate over a region rather than differentiate at a point, start with the double integral calculator.

Key terms

Partial derivative

The derivative of a multivariable function with respect to one variable, with all other variables held constant. Written ∂f/∂x or f_x.

Mixed partial

A second derivative taken in two different variables, such as f_xy. Equal to f_yx wherever both are continuous.

Gradient

The vector (f_x, f_y, f_z). It points in the direction of fastest increase and its length is the largest directional derivative at the point.

Hessian

The matrix of all second partials. Its determinant in two variables is the discriminant D = f_xxf_yy − f_xy², used to classify critical points.

Clairaut's theorem

If the second partials of f are continuous on an open set, then f_xy = f_yx there. Also known as Schwarz's theorem.