How do I take a partial derivative with respect to one variable?

Type the expression and write that variable on its own in the variables box. Every other symbol is treated as a constant, and the page lists which symbols those were beside the answer. For x^2*y+sin(y) differentiated by x the answer is 2*x*y, with y held constant, and the page also shows the y partial beside it for comparison.

Does the order of differentiation matter here?

The notation is read left to right in the order applied: ∂²f/∂x∂y means x first and then y. For smooth functions these mixed derivatives agree. The page checks both orders and marks a contradictory result Unresolved.

How many derivatives can I take at once?

Up to 5 single differentiations per question. Times each repeats every listed variable: x, y at 2 means x, x, y, y, while x, y, z at 2 asks for six steps and is refused. Taylor mode supports degrees 0 through 10 and calculates one extra coefficient. The symbolic worker stops after 2 seconds if it has not finished.

What can I type into the expression?

Numbers, single letters for variables, the constants pi and e, the signs plus, minus, times, divide and caret, brackets, and the functions sin, cos, tan, sec, csc, cot, asin, acos, atan, sinh, cosh, tanh, exp, sqrt, abs and log. ln is read as log, which is the natural logarithm here. Multiplication can be left out where it is obvious, so 2x and 3 sin(x) read the way they are written. Expressions are read up to 500 characters, and anything outside that list is named as unsupported rather than dropped from the expression.

Can it do implicit differentiation?

Yes. Choose Implicit and write the relation as F or as an equation with one equals sign in it, then name the independent variable first and the dependent one second. The page returns dy/dx as minus the x partial over the y partial, so x^2+y^2-1 gives -x/y, and it names the points where the formula is not defined, which for the circle is where 2*y = 0. The formula is also checked against the curve itself: the page solves F = 0 for y at a point, moves x a little, solves again, and compares the slope it measured with the formula.

What does the Taylor mode print?

The polynomial sum of f to the k at the centre times x minus the centre to the k over k factorial, term by term, so exp(x) about 0 to order 3 is 1+x+x^2/2+x^3/6. Beside it the page shows the first term it left out, which for that expansion is x^4/24. The expansion is checked by comparing it against the function near the centre, and the page makes no claim about where the full series converges.

Why does it sometimes say Unresolved or Unverified?

Unresolved means a numeric check contradicted the symbolic answer, so the affected result is withheld. For (x*y)^(1/2), the symbolic method returns 1/(8*x^(1/2)*y^(3/4)) in one mixed order and 1/(8*x^(3/4)*y^(1/2)) in the other; the disagreement is shown. Unverified means a check could not run, often outside the real domain, so the answer is labelled unchecked. Checks at sampled points do not prove a formula valid at every point.

Does anything I type leave the browser?

No. The expression is read on the page, the algebra runs in a worker on the same tab from a library file this site serves, and nothing is uploaded, logged or sent to analytics. The only thing kept is your last expression and settings, in this browser's own storage, so the page opens where you left it.

How do I calculate second, third or mixed partial derivatives?

For repeated derivatives with respect to x, enter x in Variables, in order and choose 2 or 3 in Times each. For a mixed second derivative, enter x, y and choose 1. Intermediate derivatives and held symbols appear in the result. Expressions may contain three or more single-letter variables, subject to the five-step limit.

Does it handle the chain rule or show worked differentiation rules?

Supported composed expressions such as sin(x*y) can be differentiated directly. The page shows intermediate derivative results, not a rule-by-rule derivation. It does not accept separate dependency definitions such as x(t) and y(t), solve critical points, plot graphs or evaluate a derivative at a supplied point.

Can I download a result or install this as an app?

The tool provides text output and Copy. It does not generate a PDF, ZIP or downloadable result file, and this page does not provide an app or APK installer. The calculation requires a browser that permits the background worker.

Partial Derivative Calculator

Enter an expression and choose Partial, Implicit or Taylor. Partial mode shows derivatives in the variable order you choose, holding other symbols constant. Implicit mode finds a first derivative from a relation, and Taylor mode builds a finite polynomial about a chosen centre. Results include numeric checks and a Copy control.

expressionx, then y

f = x^2*y+sin(y) variables      x, then yorder          1 on each, 2 in allheld constant  none, every symbol here is differentiated at some step

derivativesworking

Working out ∂²f/∂x∂y. Working out ∂²f/∂x∂y.

Mode

Expression

Variables, in order

Times each

∂f/∂x: working
∂²f/∂x∂y: working
Order: x, then y
Result: Working

Working out ∂²f/∂x∂y.

∂/∂x of x²y + sin(y)y held constant
2*x*y
Then ∂/∂y of thatmixed second order, asked both ways round
2*x
x² + y² - 1 = 0, implicitdy/dx = -Fx/Fy, singular where 2*y = 0
-x/y

Accuracy. Symbolic differentiation in a sandboxed worker, exact, with the differentiation order printed beside the answer because the order you pick is part of the question. Every symbol you did not list is treated as a constant and the page says which those were. A function outside the supported set is named as unsupported rather than quietly dropped from the expression.

How this is worked out

The algebra is nerdamer 1.1.13, running in a worker of its own with a 2 second clock beside it: some expressions send it a long way, and the page stops it rather than freezing with it. Each step is a question of its own, nested inside the step before it, so the second partial is the derivative of the first and is checked against it rather than against the expression you typed.

Nothing the library returns is printed until it has been checked here on the page. Every derivative is compared against a slope measured on the expression that went into it, by central difference with one Richardson step, at up to 6 points, and they have to agree to within 0.0001 of the larger of one and the value. A mixed second partial is asked for a second time in the opposite order and the two answers are compared. An implicit result is checked against the curve itself: the page solves the relation for the dependent variable at a point, moves the independent one a little, solves again, and measures the slope between them. A Taylor expansion has to match the function at the centre and its error has to fall by the factor its order requires when the distance from the centre is halved. When a check contradicts the algebra, the page prints neither and shows what each one said.

What you can type

Numbers, single letters for variables, the constants pi and e, the signs + - * / ^ and brackets, and these functions: sin, cos, tan, sec, csc, cot, asin, acos, atan, sinh, cosh, tanh, exp, sqrt, abs, log. ln is read as log, which is the natural logarithm here, and arcsin, arccos and arctan are read as asin, acos and atan. Multiplication can be left out where it is obvious, so 2x and 3 sin(x) read the way they are written. Expressions are read up to 500 characters, a question may ask for up to 5 differentiations, and a Taylor expansion goes to order 10. A function outside that list is named as unsupported rather than dropped from the expression.

Common questions

How do I take a partial derivative with respect to one variable?: Type the expression and write that variable on its own in the variables box. Every other symbol is treated as a constant, and the page lists which symbols those were beside the answer. For x^2*y+sin(y) differentiated by x the answer is 2*x*y, with y held constant, and the page also shows the y partial beside it for comparison.
Does the order of differentiation matter here?: The notation is read left to right in the order applied: ∂²f/∂x∂y means x first and then y. For smooth functions these mixed derivatives agree. The page checks both orders and marks a contradictory result Unresolved.
How many derivatives can I take at once?: Up to 5 single differentiations per question. Times each repeats every listed variable: x, y at 2 means x, x, y, y, while x, y, z at 2 asks for six steps and is refused. Taylor mode supports degrees 0 through 10 and calculates one extra coefficient. The symbolic worker stops after 2 seconds if it has not finished.
What can I type into the expression?: Numbers, single letters for variables, the constants pi and e, the signs plus, minus, times, divide and caret, brackets, and the functions sin, cos, tan, sec, csc, cot, asin, acos, atan, sinh, cosh, tanh, exp, sqrt, abs and log. ln is read as log, which is the natural logarithm here. Multiplication can be left out where it is obvious, so 2x and 3 sin(x) read the way they are written. Expressions are read up to 500 characters, and anything outside that list is named as unsupported rather than dropped from the expression.
Can it do implicit differentiation?: Yes. Choose Implicit and write the relation as F or as an equation with one equals sign in it, then name the independent variable first and the dependent one second. The page returns dy/dx as minus the x partial over the y partial, so x^2+y^2-1 gives -x/y, and it names the points where the formula is not defined, which for the circle is where 2*y = 0. The formula is also checked against the curve itself: the page solves F = 0 for y at a point, moves x a little, solves again, and compares the slope it measured with the formula.
What does the Taylor mode print?: The polynomial sum of f to the k at the centre times x minus the centre to the k over k factorial, term by term, so exp(x) about 0 to order 3 is 1+x+x^2/2+x^3/6. Beside it the page shows the first term it left out, which for that expansion is x^4/24. The expansion is checked by comparing it against the function near the centre, and the page makes no claim about where the full series converges.
Why does it sometimes say Unresolved or Unverified?: Unresolved means a numeric check contradicted the symbolic answer, so the affected result is withheld. For (x*y)^(1/2), the symbolic method returns 1/(8*x^(1/2)*y^(3/4)) in one mixed order and 1/(8*x^(3/4)*y^(1/2)) in the other; the disagreement is shown. Unverified means a check could not run, often outside the real domain, so the answer is labelled unchecked. Checks at sampled points do not prove a formula valid at every point.
Does anything I type leave the browser?: No. The expression is read on the page, the algebra runs in a worker on the same tab from a library file this site serves, and nothing is uploaded, logged or sent to analytics. The only thing kept is your last expression and settings, in this browser's own storage, so the page opens where you left it.
How do I calculate second, third or mixed partial derivatives?: For repeated derivatives with respect to x, enter x in Variables, in order and choose 2 or 3 in Times each. For a mixed second derivative, enter x, y and choose 1. Intermediate derivatives and held symbols appear in the result. Expressions may contain three or more single-letter variables, subject to the five-step limit.
Does it handle the chain rule or show worked differentiation rules?: Supported composed expressions such as sin(x*y) can be differentiated directly. The page shows intermediate derivative results, not a rule-by-rule derivation. It does not accept separate dependency definitions such as x(t) and y(t), solve critical points, plot graphs or evaluate a derivative at a supplied point.
Can I download a result or install this as an app?: The tool provides text output and Copy. It does not generate a PDF, ZIP or downloadable result file, and this page does not provide an app or APK installer. The calculation requires a browser that permits the background worker.

Symbolic differentiation in a sandboxed worker, exact, with the differentiation order printed beside the answer because the order you pick is part of the question. Every symbol you did not list is treated as a constant and the page says which those were. A function outside the supported set is named as unsupported rather than quietly dropped from the expression.