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This partial derivative calculator differentiates functions of two or three variables, such as \(f(x, y) = x^2y + \sin(xy)\). Choose the variable (x, y, or z), and it shows every step. All the other letters are treated as constants, just like your textbook does.
How to Use It
- Type a function with more than one variable, for example
x^2 y + sin(x y). Put a space or*between letters that multiply. - Choose which variable to differentiate with respect to.
- Pick the order. Order 2 gives \(\frac{\partial^2 f}{\partial x^2}\).
- Press Solve.
What Is a Partial Derivative?
When a function depends on more than one input, you can ask how it changes when only one input changes. That is a partial derivative.
The symbol is a curly d: \(\frac{\partial f}{\partial x}\). It is read “partial f partial x.” Other ways to write it are \(f_x\) and \(\partial_x f\).
The rule is simple: to find \(\frac{\partial f}{\partial x}\), pretend every other variable is a number, then use the normal derivative rules.
Worked Examples
Example 1: \(f(x,y) = x^2y + 3y^2\)
With respect to x: treat y as a constant. \(\frac{\partial f}{\partial x} = 2xy + 0 = 2xy\).
With respect to y: treat x as a constant. \(\frac{\partial f}{\partial y} = x^2 + 6y\).
Example 2: \(f(x,y) = e^{xy}\)
Use the chain rule. The inside is \(xy\).
\(\frac{\partial f}{\partial x} = y\,e^{xy}\) and \(\frac{\partial f}{\partial y} = x\,e^{xy}\).
Example 3: \(f(x,y) = \ln(x^2 + y^2)\)
\(\frac{\partial f}{\partial y} = \frac{2y}{x^2 + y^2}\).
Example 4: \(f(x,y,z) = xyz + z^2\)
\(\frac{\partial f}{\partial z} = xy + 2z\).
Second-Order and Mixed Partials
You can take partial derivatives more than once:
- \(f_{xx} = \frac{\partial^2 f}{\partial x^2}\): differentiate with respect to x twice.
- \(f_{xy}\): first with respect to x, then with respect to y.
For most functions you meet in class, \(f_{xy} = f_{yx}\). This is called Clairaut’s theorem. To find a mixed partial here, solve for \(f_x\) first, then paste that answer back in and choose y.
Where Partial Derivatives Are Used
- Gradients: the vector \(\nabla f = \left(f_x, f_y\right)\) points uphill on a surface.
- Tangent planes and linear approximation in multivariable calculus.
- Optimization: set \(f_x = 0\) and \(f_y = 0\) to find high and low points.
- Physics and economics: heat flow, waves, and how cost changes with one input.
Common Mistakes
- Differentiating the other variable. For \(\frac{\partial}{\partial x}(x^2y)\), the y stays. The answer is \(2xy\), not \(2x\).
- Treating a constant term wrong. \(\frac{\partial}{\partial x}(3y^2) = 0\), because it has no x.
- Typing xy as one name. Write
x yorx*yso the calculator sees two variables.
Learn the Math Behind the Partial Derivative Calculator
A partial derivative changes one variable and holds the others fixed. Partial derivatives build gradients, Jacobians and optimization methods.
Frequently Asked Questions
How many variables can I use?
You can use x, y, and z, plus constants like a, b, or k. Any letter you do not differentiate is treated as a constant.
Can it find integrals of functions with two variables?
Yes. Use the double integral calculator or the triple integral calculator.
Why is there no graph?
A function of two variables is a 3D surface. The graph tab is for one-variable functions. The steps show everything you need.