Partial Derivative Calculator

A partial derivative measures how a function of several variables changes as you vary one variable while holding the others constant. For a function f(x, y), the partial derivative with respect to x treats y as a fixed number and differentiates as usual. It is written with the rounded partial symbol, for example the partial of f with respect to x.

Type the function using * for multiplication and ^ for powers, so x squared times y becomes x^2*y. Variables can be any letters, and you can use functions like sin, cos, exp, ln, log, and sqrt. The constants pi and e are recognized, and parentheses group terms the way you expect.

Pick an order from 1 to 4 to differentiate the same variable repeatedly, such as the second partial with respect to x. To get a mixed partial, turn on the mixed option and choose a second variable for the last step, for example differentiating by x and then by y. For smooth functions the mixed partials are equal regardless of order, a result known as Clairaut's theorem.

That is the definition of a partial derivative. When you differentiate with respect to x, every other variable is treated as a constant, so its own derivative is zero. This isolates the rate of change in one direction, which is exactly what the gradient and many physics and economics formulas need.

Yes. Fill in a value for every variable in the evaluate section, and the calculator shows the numeric value of both the function and its partial derivative at that point, rounded to four decimals. If a value falls outside the function's domain, such as ln of a negative number, the numeric result is left blank.

It applies the exact rules of calculus, including the power, product, quotient, and chain rules, along with the derivatives of sin, cos, tan, exp, ln, log base 10, sqrt, and the inverse and hyperbolic trig functions. Results are symbolic and exact, not numerical approximations. Very long or unusual expressions may not collapse to the most compact form, but the answer stays mathematically correct.