Partial Derivative Calculator

One math box plus one dropdown, pre-filled with x^2 - 3xy + 2y^2 - 4x + 5*y - 12 and d/dx. Click Calculate. This partial derivative calculator asks for a function of two variables and the variable to differentiate against. That is the entire partial derivative calculator setup.

• The function box already holds real math: type x^2 - 3xy + 2*y^2 and see fractions, exponents, and products typeset as you go.
• The variable dropdown picks the direction: x or y. Everything else is held constant automatically.
• One answer comes back: the partial derivative, as an expression in x and y.

This partial derivative calculator keeps results exact, so the default returns 2x − 3y − 4 with no rounding anywhere.

This page is more than a static partial derivative calculator — it is also an AI tutor you can ask questions:

• Ask in plain words. "Partial derivative of x squared plus 3 y squared with respect to y." The AI fills the form and runs this partial derivative calculator for you.
• Take a photo of your homework. The AI reads the function and the target variable, then fills the partial derivative calculator for you.
• Ask about the ideas, not just the answer. "Why does the −3xy term turn into −3y?" The AI explains step by step — every expression still comes from the partial derivative calculator, so nothing is guessed.
• Keep asking follow-ups. Ask "why?" as often as you need — follow-up questions are welcome.
• Second derivatives and three variables. The form does clean first-order partials; the AI on this page walks mixed and higher-order partials with you.

A function of two variables has no single slope — it has one per direction. The partial derivative calculator exists for exactly this. The partial derivative calculator computes them one at a time: treat every other variable as a constant, then differentiate with the ordinary rules you already know. Freeze y, and ∂f/∂x reads the surface along the x direction; freeze x, and ∂f/∂y reads it along y. The partial derivative calculator does the freezing for you.

The picture shows both slices of z = x² + 3y² through the same point (1, 4). The y = 1 slice climbs with slope 2 — that is ∂z/∂x at the point. The x = 1 slice is steeper, slope 6 — that is ∂z/∂y. Same point, two slopes: exactly what the partial derivative calculator reports, one slice at a time — the partial derivative calculator's whole job in one picture.

Read the picture first: one slice, one slope. Now the symbols the partial derivative calculator uses:

$\frac{\partial f}{\partial x} = \lim_{h \to 0} \frac{f(x+h,\ \y) - f(x,\ \y)}{h}$

Every answer the partial derivative calculator returns is one slice slope written in symbols. That is all a partial derivative calculator is: one direction at a time. Ordinary rules — power, product, chain — all still apply inside each slice.

Find both first-order partials of f(x, y) = x² − 3xy + 2y² − 4x + 5y − 12.

1. Freeze y: the terms −3xy and 2y² act like constants. $\partial f/\partial x = 2x - 3y - 4$.
2. Freeze x: $\partial f/\partial y = -3x + 4y + 5$.
3. Check the cross terms: −3xy leaves −3y in the first and −3x in the second — the product rule at work.

This exact function comes pre-filled in the partial derivative calculator above. Switch the dropdown from x to y and the partial derivative calculator swaps the answer instantly — two partial derivative calculator runs, two slices.

Find both partials of z = x² − 3xy + y².

1. Freeze y: $\partial z/\partial x = 2x - 3y$.
2. Freeze x: $\partial z/\partial y = -3x + 2y$.
3. Symmetry check: the mixed term −3xy hands each partial the same −3 coefficient with the other variable attached.

Type x^2 - 3xy + y^2 into the partial derivative calculator above and both partials confirm, one dropdown flip apart. The partial derivative calculator never drops the −3 partner.

The old rules never left. Take f = sin(x)·cos(y) and g = e^(xy).

1. $\partial f/\partial x = \cos x \cos y$ — cos(y) rides along as a constant coefficient.
2. $\partial g/\partial x = y\,e^{xy}$ — the chain rule with y frozen, exactly as in single-variable calculus.
3. Flip the variable and the constant re-appears: $\partial g/\partial y = x\,e^{xy}$.

Feed sin(x)cos(y) to the partial derivative calculator above — it returns cos(x) cos(y) in one click, chain rule included.

Four slips the partial derivative calculator catches every day:

1. Freezing the wrong variable. Differentiating with respect to x means x moves and y is the constant — backwards, and the partial derivative calculator's answer is the other slice.
2. Dropping the partner on product terms. ∂/∂x of −3xy is −3y, not 0 and not −3: x still varies, so the product rule survives with y as the constant. The partial derivative calculator never forgets the partner.
3. Writing d instead of ∂. The curly ∂ flags a multivariable derivative along one direction; plain d is for single-variable work. Same family, different meaning.
4. Reporting an answer with no direction. "The derivative is 2x − 3y − 4" is unfinished: with respect to x, or to y? Label every partial derivative calculator result with its variable — an unlabeled answer is the classic partial derivative calculator misuse.