Partial Derivative Calculator

The partial derivative calculator differentiates a function of two variables, \( f(x, y) \), with respect to each variable. It returns the first partials \( f_x \) and \( f_y \) as formulas and as values at a point you choose, plus the second partials \( f_{xx} \), \( f_{yy} \) and \( f_{xy} \).

It’s made for multivariable calculus students checking their work on homework, tangent planes and critical-point problems. It’s free and needs no sign-up.

How to use the partial derivative calculator

1. Type \( f(x, y) \) using x and y. Use ^ for powers and sqrt(), sin(), cos(), ln(), e^(x y) and pi for functions. Implicit multiplication works, so x^2 y means \( x^2 \cdot y \) and sin(x y) means \( \sin(xy) \).
2. Enter the point \( (x_0, y_0) \) where you want the values.
3. Press Differentiate. The top rows show \( f_x \) and \( f_y \), each as a formula and its value at your point.
4. Read the second partials \( f_{xx} \), \( f_{yy} \) and \( f_{xy} \) in the rows below. These are given as formulas, so substitute the point yourself if you need their values.

How it works

A partial derivative measures how \( f \) changes when only one variable moves:

$$f_x = \frac{\partial f}{\partial x} = \lim_{h\to0}\frac{f(x+h, y) - f(x, y)}{h}$$

and similarly for \( f_y \), with \( h \) added to \( y \). In practice you treat the other variable as a constant and use the ordinary rules: power, product, quotient and chain rule.

The calculator does this symbolically. It differentiates with respect to \( x \) while holding \( y \) fixed, then with respect to \( y \) while holding \( x \) fixed, simplifies, and evaluates at your point. The second partials are the partials of the partials:

$$f_{xx} = \frac{\partial^2 f}{\partial x^2}, \quad f_{yy} = \frac{\partial^2 f}{\partial y^2}, \quad f_{xy} = \frac{\partial}{\partial y}\big(f_x\big)$$

For the smooth functions you meet in a calculus course, the mixed partials agree (Clairaut’s theorem), so \( f_{xy} = f_{yx} \) and one row covers both.

Worked example

Example 1: a polynomial. Let \( f(x, y) = x^3y^2 - 4xy \). Find all first and second partials, and the first partials at \( (1, 2) \).

Hold \( y \) constant and differentiate in \( x \):

$$f_x = 3x^2y^2 - 4y$$

Hold \( x \) constant and differentiate in \( y \):

$$f_y = 2x^3y - 4x$$

At \( (1, 2) \): \( f_x = 3(1)(4) - 8 = 4 \) and \( f_y = 2(1)(2) - 4 = 0 \). So near this point \( f \) rises at a rate of 4 in the \( x \)-direction and is momentarily flat in the \( y \)-direction.

Second partials: \( f_{xx} = 6xy^2 \), \( f_{yy} = 2x^3 \), and \( f_{xy} = 6x^2y - 4 \). As a check, differentiating \( f_y \) in \( x \) also gives \( 6x^2y - 4 \). At \( (1, 2) \) these are 24, 2 and 8. The calculator above is preloaded with this function and point.

Example 2: the chain rule. Let \( g(x, y) = \sin(x^2y) \) at \( (1, \pi) \). The inner function is \( x^2 y \), so

$$g_x = 2xy\cos(x^2y), \qquad g_y = x^2\cos(x^2y).$$

At \( (1, \pi) \), \( \cos\pi = -1 \), so \( g_x = -2\pi \approx -6.2832 \) and \( g_y = -1 \). Type sin(x^2 y) with \( x_0 = 1 \), \( y_0 = \) pi to compare.

Common mistakes

• Differentiating the “constant” variable. In \( \frac{\partial}{\partial x}(4xy) \), \( y \) is a constant factor, so the answer is \( 4y \), not \( 4y + 4x \).
• Dropping a factor with the chain rule. For \( \sin(x^2y) \), the \( x \)-partial needs the inner derivative \( 2xy \), and the \( y \)-partial needs \( x^2 \). This is the single-variable chain rule with one variable frozen.
• Mixing up \( f_{xy} \) notation. \( f_{xy} \) means “first \( x \), then \( y \)”, while \( \frac{\partial^2 f}{\partial x\,\partial y} \) is read right to left. For smooth functions they match, so the order rarely matters for the answer.
• Stopping at \( f_x \) and \( f_y \). Critical points need both first partials equal to 0, and classifying them needs \( f_{xx} \), \( f_{yy} \) and \( f_{xy} \) for the second-derivative test.

For more worked examples and the geometric meaning of each partial, read the guide to partial derivatives. The two first partials together form the gradient, covered in gradient and directional derivative; the gradient calculator evaluates it at a point along with directional derivatives.

Worked example: all the partial derivatives

For \( f(x, y) = x^2y + 3xy^2 \):

• \( f_x = 2xy + 3y^2 \) (treat \( y \) as a constant)
• \( f_y = x^2 + 6xy \) (treat \( x \) as a constant)
• At \( (1, 2) \): \( f_x = 4 + 12 = 16 \) and \( f_y = 1 + 12 = 13 \).
• Mixed partials: \( f_{xy} = 2x + 6y = f_{yx} \). For smooth functions the order does not matter (Clairaut’s theorem).

The tangent plane

Partial derivatives give the plane that touches a surface at a point:

$$z = f(a, b) + f_x(a, b)(x - a) + f_y(a, b)(y - b)$$

At \( (1, 2) \), \( f = 2 + 12 = 14 \), so the tangent plane is \( z = 14 + 16(x - 1) + 13(y - 2) \).

Where partial derivatives are used

• Economics: the marginal effect of one input (labour) while others (capital) stay fixed.
• Physics: heat and wave equations are written with partial derivatives.
• Machine learning: training a model follows the gradient, the vector of partial derivatives.

Practice problems

1. \( f(x, y) = e^{xy} \): find \( f_x \) and \( f_y \).
2. \( f(x, y) = \sin(x)\cos(y) \): find \( f_{xy} \).

Answers: 1. \( f_x = ye^{xy} \), \( f_y = xe^{xy} \). 2. \( -\cos(x)\sin(y) \).

Understanding the Calculator's Output

The partial derivative calculator presents results in a structured way. First, it shows the formulas for the first partial derivatives, f_x and f_y. These formulas describe the rate of change across the entire function domain. Below each formula, you will see its numerical value at the specific point you entered. This value tells you the exact slope of the surface in that direction at that point.

Next, the calculator lists the second partial derivatives: f_xx, f_yy, and f_xy. These are also given as formulas. These second derivatives are crucial for analyzing the curvature of the surface. For instance, f_xx indicates how the slope in the x-direction changes as x varies. The mixed partial f_xy shows how the x-slope changes with respect to y, or vice versa.

Tips for Complex Functions

Inputting complex functions correctly ensures accurate results from any partial derivative calculator. Always use parentheses to clarify the order of operations, especially in denominators or exponents. For example, type (x+y)^2 instead of x+y^2. Be careful with trigonometric functions like sin(xy) versus sin(x)*y; the calculator interprets implicit multiplication.

Break down very long functions into smaller parts if you are unsure about the syntax. You can then compute partials for each part and combine them. Double-check your variable names; ensure you use 'x' and 'y' consistently as expected by the tool. A small typo can lead to an error or an incorrect result.

Related guides

• Partial Derivatives: How to Calculate Them
• Gradient and Directional Derivative: Formulas and Examples
• Chain Rule Explained: Formula, Steps and 8 Examples
• Double Integrals: How to Evaluate Them Step by Step

Further reading

• Partial Derivatives (Paul’s Online Math Notes) — notation, the rules and implicit differentiation with several variables.
• Higher Order Partial Derivatives (Paul’s Online Math Notes) — second partials and Clairaut’s theorem on mixed partials.

FAQ

Does the partial derivative calculator show second derivatives?

Yes. It gives \( f_{xx} \), \( f_{yy} \) and the mixed partial \( f_{xy} \) as formulas, below the first partials.

Can it handle functions of three variables?

No. This calculator works with \( f(x, y) \). For \( f(x, y, z) \), treat two variables as constants and differentiate by hand, or fix \( z \) at a number and use the calculator for the other two.

What does a partial derivative of zero mean?

The function is momentarily flat in that direction. If both \( f_x \) and \( f_y \) are zero, the point is a critical point: a local maximum, minimum or saddle.

Is the partial derivative calculator free?

Yes. It works in your browser with no sign-up and no limits.