Higher-order partial derivatives

Mixed Partial Derivatives: fxy vs fyx and Differentiation Order

To find fxx, fxy, fyx and fyy, draw two first-derivative branches from the original function. Differentiate each branch once more. This separates repeated differentiation in one direction from mixed differentiation. The last letter in this page’s subscript is the second operation; in an operator fraction the rightmost operator acts first. Writing “first x, then y” beside the symbols prevents an order error when comparing books or software.

Direct answer

f_{xy}=\frac{\partial}{\partial y}\left(\frac{\partial f}{\partial x}\right)

Check the notation convention before calculating. Continuous second partial derivatives in a neighborhood are a sufficient condition for f_xy=f_yx; equality is not unconditional.

Four second partials from two first-derivative branches

Take f(x,y)=x²y+y³. For f_x, hold y fixed: the first term becomes 2xy and the y³ term has derivative zero. For f_y, return to the original function: x²y becomes x² and y³ becomes 3y². Thus f_x=2xy and f_y=x²+3y² are independent answers, not consecutive steps.

Starting from f_x=2xy, differentiation with respect to x gives f_xx=2y. Differentiation with respect to y instead gives f_xy=2x. Do not add a second derivative to the original expression and label it f_xy: the first result is the input to the second operation.

Starting from f_y=x²+3y², differentiation with respect to x gives f_yx=2x. Differentiation with respect to y gives f_yy=6y. The two mixed derivatives agree here because the polynomial has continuous partial derivatives of every order.

Schwarz’s theorem gives a useful sufficient condition: if both second mixed partial derivatives are continuous in a neighborhood of the point, their values agree there. Failure to meet the condition does not by itself prove inequality. Conversely, checking one polynomial does not prove the rule for every function, especially at exceptional points of piecewise functions.

In the calculator, use the independent-variable selection when you want f_x and f_y from the original function. To follow the mixed path, calculate only f_x first, then choose “continue with y” below that successful result. Keep the intermediate expression and active variable visible. Re-enter the original function before computing a separate first derivative.

f(x,y)=x^2y+y^3

f_x=2xy,\quad f_y=x^2+3y^2

f_{xx}=2y,\quad f_{xy}=2x

f_{yx}=2x,\quad f_{yy}=6y

Worked examples

Worked examples
Task	Answer	Conditions
Original f → x	$f_x=2xy$	Hold y fixed; start at the original function.
Original f → y	$f_y=x^2+3y^2$	Return to the original function, rather than differentiating f_x.
f_x → x	$f_{xx}=2y$	A pure second derivative repeats the same variable.
f_x → y	$f_{xy}=2x$	First x, then y: the intermediate input is 2xy.
f_y → x	$f_{yx}=2x$	First y, then x: the intermediate input is x²+3y².
f_y → y	$f_{yy}=6y$	The original x² term is constant in this second y derivative.

Common mistakes

• Reporting f_y from the original function as the next step after f_x: independent partials are not a mixed derivative.
• Reading only a subscript without checking the author’s order convention. Write both operators or the words “first x, then y”.
• Assuming mixed derivatives are always equal, or claiming they differ merely because a sufficient continuity condition has not been verified.

Check the four paths on a different polynomial

Use g(x,y)=x³y². Find the independent first derivatives before choosing a second direction.

g(x,y)=x³y²: find g_x and g_y independently

Check before solving

Both operations start from g.

Reveal the reference answer

Reference answer

g_x=3x²y²; g_y=2x³y

Find g_xx, g_xy, g_yx and g_yy

Check before solving

Keep each intermediate branch.

Reveal the reference answer

Reference answer

g_xx=6xy²; g_xy=6x²y; g_yx=6x²y; g_yy=2x³

Try the worked examples in the calculator

Each link opens the matching calculator with the expression, variable and required conditions already filled in.

These links load the original function with x or y selected and do not submit a calculation. For f_xy, calculate the x example, then continue with y from its result. Each submitted derivative follows the existing calculation and credit rules.

Partial derivative calculator:

x^{2}\,y + y^{3}

Partial derivative calculator:

x^{2}\,y + y^{3}

Open the calculator without a preset: Partial derivative calculator

References

OpenStax Calculus Volume 3, 4.3: Partial Derivatives and Higher-Order Derivatives

Frequently asked questions

Is finding f_x and f_y the same as finding f_xy?+

No. The first task differentiates the original function twice independently. The second differentiates f_x with respect to y.

Which derivative is taken first in f_xy?+

On this page, x comes first and y second. The equivalent operator expression is ∂/∂y(∂f/∂x). State the convention because notation is not universal.

Must f_xy equal f_yx?+

Not for every function at every point. Continuous second partials in a neighborhood are a standard sufficient condition. Check exceptional points separately.

Conclusion

For mixed partial derivatives, preserve the original function, identify the first direction and keep its result before taking the second direction. Use separate branches for independent partials and check the hypotheses before interchanging the order.

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