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Calculus Solver

Partial Derivative Calculator

Differentiate a function of two or more variables with respect to one of them while holding the others constant. Get first partials such as ∂f/∂x and second partials such as fxx or the mixed partial fxy, with every rule shown.

Your expression will be shown here as it will be read.

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What is a partial derivative?

A function such as f(x, y) describes a surface. A partial derivative measures how f changes when only one variable moves while the others stay fixed. ∂f/∂x is the slope of the surface in the x-direction, holding y constant; ∂f/∂y is the slope in the y-direction, holding x constant.

In practice, you compute a partial derivative with the ordinary rules of differentiation, treating every other variable as if it were a number.

∂f∂x=lim⁡h→0f(x+h,y)−f(x,y)h\frac{\partial f}{\partial x} = \lim_{h \to 0} \frac{f(x + h, y) - f(x, y)}{h}

Rules and notation for partial derivatives

Nothing about the derivative rules changes; what changes is which letters count as constants.

  • Notation

    ∂f∂x=fx,∂f∂y=fy\frac{\partial f}{\partial x} = f_{x},\qquad \frac{\partial f}{\partial y} = f_{y}

    The curly ∂ signals that other variables are held constant.

  • Hold the others constant

    ∂∂x[x2y3]=2x y3\frac{\partial}{\partial x}\left[x^{2}y^{3}\right] = 2x\,y^{3}

    y³ behaves like a constant factor, exactly like the 5 in 5x².

  • Same rules as usual

    ∂∂x[exy]=y exy\frac{\partial}{\partial x}\left[e^{xy}\right] = y\,e^{xy}

    The chain rule still applies. Here the inner function xy has ∂/∂x = y.

  • Second partials

    fxx=∂2f∂x2,fxy=∂2f∂y ∂xf_{xx} = \frac{\partial^{2} f}{\partial x^{2}},\qquad f_{xy} = \frac{\partial^{2} f}{\partial y\,\partial x}

    fxy means: differentiate with respect to x first, then y.

  • Clairaut's theorem

    fxy=fyxf_{xy} = f_{yx}

    If both mixed partials are continuous, the order of differentiation does not matter.

  • Gradient

    ∇f=(fx, fy)\nabla f = \left(f_{x},\ f_{y}\right)

    The vector of first partials points in the direction of steepest increase.

How this calculator works

The calculator uses the same rule-based differentiator as the derivative calculator, with ∂ notation and one change: every letter other than the one you choose is treated as a constant. Each rule it applies is shown as a step.

For second partials, choose a second variable. The calculator differentiates with respect to the first variable, simplifies, then differentiates with respect to the second, showing both stages.

Each stage is checked numerically: the other variables are fixed at test values, and the result is compared with a finite-difference estimate in the chosen direction. Supported variable names are x, y, z, t, u and θ.

For the full details, including how answers are verified, see how the solver works.

Worked examples

Work through these by hand first, then enter them above to compare each step.

Example 1

Partial derivative with respect to x

∂∂x[x2y+3xy3]\frac{\partial}{\partial x}\left[x^{2}y + 3x y^{3}\right]

  1. Treat y as a constant. In x²y, the factor y stays; in 3xy³, the factor 3y³ stays.

    2x⋅y+3y3⋅12x\cdot y + 3y^{3}\cdot 1

Answer

fx=2xy+3y3f_{x} = 2xy + 3y^{3}

Example 2

Partial derivative with respect to y

∂∂y[x2y+3xy3]\frac{\partial}{\partial y}\left[x^{2}y + 3x y^{3}\right]

  1. Now x is the constant. x²·y gives x², and 3x·y³ gives 3x·3y².

    x2⋅1+3x⋅3y2x^{2}\cdot 1 + 3x\cdot 3y^{2}

Answer

fy=x2+9xy2f_{y} = x^{2} + 9xy^{2}

Example 3

Mixed second partial

fxy for f=x2y+3xy3f_{xy}\ \text{for}\ f = x^{2}y + 3x y^{3}

  1. Start from fx = 2xy + 3y³ and differentiate with respect to y:

    ∂∂y[2xy+3y3]=2x+9y2\frac{\partial}{\partial y}\left[2xy + 3y^{3}\right] = 2x + 9y^{2}

  2. Differentiating fy = x² + 9xy² with respect to x gives the same 2x + 9y², as Clairaut's theorem predicts.

Answer

fxy=2x+9y2f_{xy} = 2x + 9y^{2}

Example 4

Chain rule inside a partial

∂∂x exy2\frac{\partial}{\partial x}\,e^{x y^{2}}

  1. The outer function is eᵘ with u = xy². Holding y constant, ∂u/∂x = y².

    exy2⋅y2e^{x y^{2}}\cdot y^{2}

Answer

fx=y2exy2f_{x} = y^{2}e^{x y^{2}}

Example 5

Three variables

∂∂z[xyz+sin⁡(xz)]\frac{\partial}{\partial z}\left[xyz + \sin(xz)\right]

  1. x and y are constants. xyz gives xy; sin(xz) needs the chain rule with ∂(xz)/∂z = x.

    xy+cos⁡(xz)⋅xxy + \cos(xz)\cdot x

Answer

fz=xy+xcos⁡(xz)f_{z} = xy + x\cos(xz)

Example 6

Second partial fxx

fxx for f=ln⁡(x2+y2)f_{xx}\ \text{for}\ f = \ln\left(x^{2} + y^{2}\right)

  1. First partial:

    fx=2xx2+y2f_{x} = \frac{2x}{x^{2} + y^{2}}

  2. Quotient rule with respect to x again:

    fxx=2(x2+y2)−2x⋅2x(x2+y2)2f_{xx} = \frac{2\left(x^{2} + y^{2}\right) - 2x\cdot 2x}{\left(x^{2} + y^{2}\right)^{2}}

Answer

fxx=2y2−2x2(x2+y2)2f_{xx} = \frac{2y^{2} - 2x^{2}}{\left(x^{2} + y^{2}\right)^{2}}

Common mistakes

  • Differentiating the constant variables

    In ∂/∂x of x²y³, the y³ is a constant factor. Writing 3y² anywhere in the answer means y was differentiated by mistake.

  • Dropping the inner partial in the chain rule

    ∂/∂x of sin(xy) is y·cos(xy), not cos(xy). The inner function xy still contributes its partial derivative.

  • Reading fxy in the wrong order

    fxy means x first, then y. The results usually agree, but on tests with discontinuous functions the order matters.

  • Losing terms that don't contain the variable

    Terms with no x, such as 5y² in ∂/∂x, differentiate to 0. They do not stay in the answer.

Limitations

  • Up to second-order partials (fx, fxx, fxy and so on). Higher orders can be found by running the result through the calculator again.
  • Variables can be x, y, z, t, u or θ. Other letters are treated as constants.
  • Gradients, Hessians, directional derivatives and the multivariable chain rule are not computed as such, but you can assemble them from the partials.

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Frequently asked questions

How do you find a partial derivative?

Pick the variable to differentiate with respect to, treat every other variable as a constant, and apply the usual derivative rules. The calculator shows each rule applied.

What is the difference between ∂ and d?

d is used for functions of one variable. ∂ signals a function of several variables, where the others are held constant while you differentiate.

How do I get a mixed partial like fxy?

Choose x as the first variable and y under “Then with respect to”. The calculator differentiates with respect to x, then y, and shows both stages.

Is fxy always equal to fyx?

For functions with continuous second partial derivatives, yes; this is Clairaut's theorem. Every function in a typical course meets that condition where it is defined.

Next step

Pick one of the worked examples, solve it on paper, then enter it in the calculator and compare your steps with each rule shown. Spotting where your work differs is the fastest way to improve.

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