Find Second Partial Derivatives Calculator

Easily calculate the second partial derivatives (f_xx, f_yy, f_xy, f_yx) for a given function f(x,y) and evaluate them at a point.

Calculate Second Partial Derivatives

Enter the coefficients and exponents for a two-term polynomial function f(x,y) = A*x^ay^b + B*x^cy^d, and the point (x,y) for evaluation.

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Results

The calculator finds the second partial derivatives of f(x,y) = Ax^ay^b + Bx^cy^d using the power rule for differentiation. f_xx is ∂²f/∂x², f_yy is ∂²f/∂y², f_xy is ∂²f/∂x∂y, and f_yx is ∂²f/∂y∂x.

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Function and Derivatives Table

Values of f(x,y), f_x, f_y, f_xx, f_yy, f_xy around the evaluation point.

x	y	f(x,y)	fx(x,y)	fy(x,y)	fxx(x,y)	fyy(x,y)	fxy(x,y)
Enter values and calculate to see the table.

Chart of f_xx vs x (y fixed)

Value of f_xx as x varies around the input x, with y fixed at the input y.

What are Second Partial Derivatives?

Second Partial Derivatives are derivatives of the first partial derivatives of a multivariable function. If you have a function of two variables, say f(x,y), its first partial derivatives are ∂f/∂x (or f_x) and ∂f/∂y (or f_y). Taking the partial derivatives of these first partial derivatives gives you the second partial derivatives.

There are four second partial derivatives for a function of two variables:

• f_xx = ∂²f/∂x²: Differentiate f with respect to x, then differentiate the result with respect to x again.
• f_yy = ∂²f/∂y²: Differentiate f with respect to y, then differentiate the result with respect to y again.
• f_xy = ∂²f/∂x∂y: Differentiate f with respect to x, then differentiate the result with respect to y.
• f_yx = ∂²f/∂y∂x: Differentiate f with respect to y, then differentiate the result with respect to x.

The Second Partial Derivatives provide information about the concavity of the function’s surface and are crucial in optimization problems (finding maxima, minima, and saddle points) using the second derivative test for multivariable functions, often involving the Hessian matrix.

This Second Partial Derivatives calculator helps you find these values for polynomial-like functions.

Who should use it?

Students of multivariable calculus, engineers, physicists, economists, and anyone working with functions of multiple variables will find a Second Partial Derivatives calculator useful. It’s particularly helpful for understanding the local behavior of a function around a point.

Common Misconceptions

A common misconception is that f_xy and f_yx are always different. However, according to Clairaut’s Theorem (or Schwarz’s theorem), if the second partial derivatives are continuous in a region, then the mixed partial derivatives are equal in that region (f_xy = f_yx). Our Second Partial Derivatives calculator assumes continuity for the polynomial functions it handles, so f_xy will equal f_yx.

Second Partial Derivatives Formula and Mathematical Explanation

For a function f(x,y), the first partial derivatives are found by differentiating with respect to one variable while treating the other as constant:

f_x(x,y) = ∂f/∂x

f_y(x,y) = ∂f/∂y

The Second Partial Derivatives are then:

f_xx(x,y) = ∂/∂x (∂f/∂x) = ∂²f/∂x²

f_yy(x,y) = ∂/∂y (∂f/∂y) = ∂²f/∂y²

f_xy(x,y) = ∂/∂y (∂f/∂x) = ∂²f/∂x∂y

f_yx(x,y) = ∂/∂x (∂f/∂y) = ∂²f/∂y∂x

For our calculator’s function f(x,y) = Ax^ay^b + Bx^cy^d, using the power rule (d/dx(xⁿ) = nx^n-1):

f_x = A*a*x^a-1y^b + B*c*x^c-1y^d

f_y = A*b*x^ay^b-1 + B*d*x^cy^d-1

Then the Second Partial Derivatives are:

f_xx = A*a*(a-1)*x^a-2y^b + B*c*(c-1)*x^c-2y^d

f_yy = A*b*(b-1)*x^ay^b-2 + B*d*(d-1)*x^cy^d-2

f_xy = A*a*b*x^a-1y^b-1 + B*c*d*x^c-1y^d-1

f_yx = A*b*a*x^a-1y^b-1 + B*d*c*x^c-1y^d-1

Notice f_xy = f_yx for this type of function.

Variables Table

Variable	Meaning	Unit	Typical Range
A, B	Coefficients of the terms	Dimensionless (or units of f/(x^ay^b))	Any real number
a, b, c, d	Exponents of x and y in the terms	Dimensionless	Any real number (calculator handles integers well)
x, y	Independent variables	Depends on context	Any real number
fxx, fyy, fxy, fyx	Second Partial Derivatives	Units of f / (unit of x)² or f / (unit of y)² or f / (unit of x * unit of y)	Any real number

Practical Examples (Real-World Use Cases)

Example 1: Finding Local Extrema

Consider the function f(x,y) = 2x³y² + 3xy⁴. We want to analyze its behavior around the point (1,1).

Inputs: A=2, a=3, b=2, B=3, c=1, d=4, x=1, y=1.

Using the calculator:

f_xx = 12xy² = 12(1)(1)² = 12

f_yy = 4x³ + 36xy² = 4(1)³ + 36(1)(1)² = 4 + 36 = 40

f_xy = 12x²y + 12y³ = 12(1)²(1) + 12(1)³ = 12 + 12 = 24

At (1,1), f_xx=12, f_yy=40, f_xy=24. We would then look at the Hessian matrix determinant D = f_xxf_yy – (f_xy)² = 12*40 – 24² = 480 – 576 = -96. Since D < 0, (1,1) is a saddle point if it were a critical point (where f_x=0 and f_y=0).

Example 2: Analyzing Concavity

Let f(x,y) = -x² – y² + 5. We simplify this to f(x,y) = -1*x²y⁰ – 1*x⁰y² + 5 (constant term’s derivatives are zero). Let’s look at it as two terms: -x² and -y². Term 1: A=-1, a=2, b=0. Term 2: B=-1, c=0, d=2. Point (0,0).

Inputs: A=-1, a=2, b=0, B=-1, c=0, d=2, x=0, y=0.

f_x = -2x, f_y = -2y

f_xx = -2, f_yy = -2, f_xy = 0

At (0,0), f_xx=-2, f_yy=-2, f_xy=0. D = (-2)(-2) – 0² = 4 > 0 and f_xx < 0, indicating a local maximum at (0,0).

How to Use This Second Partial Derivatives Calculator

1. Enter Function Coefficients and Exponents: Input the values for A, a, b (for the first term Ax^ay^b) and B, c, d (for the second term Bx^cy^d). If your function has fewer terms, you can set coefficients or exponents to zero appropriately.
2. Enter Evaluation Point: Input the x and y coordinates of the point at which you want to evaluate the Second Partial Derivatives.
3. Calculate: Click the “Calculate” button.
4. Read Results: The calculator will display:
   • The symbolic form of f_xx, f_yy, and f_xy (and f_yx).
   • The numerical values of f_xx, f_yy, f_xy, and f_yx evaluated at the point (x,y) you entered.
   • A table showing values around the point.
   • A chart showing how f_xx changes with x near the point.
5. Reset: Use the “Reset” button to clear inputs to default values.
6. Copy: Use “Copy Results” to copy the main findings.

The Second Partial Derivatives calculator is designed for functions that can be represented as the sum of two terms like Ax^ay^b. More complex functions would require symbolic differentiation tools.

Key Factors That Affect Second Partial Derivatives Results

The values of the Second Partial Derivatives depend directly on:

1. The form of the function f(x,y): The coefficients (A, B) and exponents (a, b, c, d) directly determine the structure of the derivatives. Higher exponents lead to more complex derivative forms.
2. The point (x,y) of evaluation: The values of x and y plugged into the symbolic derivative expressions determine the numerical result.
3. The nature of the function (polynomial, trigonometric, etc.): Our calculator handles polynomial terms. Other functions have different differentiation rules.
4. Continuity of the function and its derivatives: If the derivatives are continuous, f_xy = f_yx, simplifying analysis (understanding derivatives).
5. Interaction between x and y terms: The presence of terms involving both x and y (like x^ay^b where a, b ≠ 0) leads to non-zero mixed partial derivatives.
6. The values of the coefficients: Larger coefficients can amplify the values of the derivatives.

Frequently Asked Questions (FAQ)

What are second partial derivatives used for?

They are primarily used in multivariable calculus to find local maxima, minima, and saddle points of functions of two or more variables (using the second derivative test and the Hessian matrix), analyze concavity, and in various fields like physics (e.g., wave equation, heat equation) and economics (e.g., optimization).

What is the Hessian matrix?

The Hessian matrix of f(x,y) is a square matrix of second-order partial derivatives:
H = [[f_xx, f_xy], [f_yx, f_yy]]. Its determinant (D = f_xxf_yy – f_xyf_yx) is used in the second derivative test.

When is f_xy equal to f_yx?

f_xy = f_yx when the second partial derivatives are continuous in the region of interest. This is known as Clairaut’s Theorem or Schwarz’s theorem on the equality of mixed partials. Our calculator deals with functions where this is true.

Can this calculator handle any function f(x,y)?

No, this specific Second Partial Derivatives calculator is designed for functions of the form f(x,y) = Ax^ay^b + Bx^cy^d. It does not handle trigonometric, exponential, logarithmic, or other types of functions, nor more than two terms directly.

How do I interpret the sign of f_xx and f_yy?

f_xx tells you about the concavity of the surface z=f(x,y) in the x-direction (holding y constant). f_yy tells you about the concavity in the y-direction (holding x constant). Positive values suggest concave up (like a valley), negative suggest concave down (like a hill) in that direction.

What if my function has only one term, or more than two?

If it has one term, set B=0, c=0, d=0. If it has more than two, you’d need a more advanced symbolic differentiator or calculate derivatives for pairs of terms separately if they are additive.

What if the exponents are not integers?

The power rule for differentiation still applies for non-integer exponents, and the calculator should handle real number exponents.

What does it mean if a second partial derivative is zero?

If f_xx=0 at a point, it suggests a possible inflection point in the x-direction. If the Hessian determinant is zero, the second derivative test is inconclusive.