Find Fxx Fyy Fxy Fyx Calculator

Calculate Second Order Partial Derivatives

Enter the coefficients and powers for a function f(x,y) of the form:
f(x,y) = c1*x^p1*y^q1 + c2*x^p2*y^q2 + c3*x^p3*y^q3 + c4*x + c5*y + c6
and the point (x, y) at which to evaluate the derivatives.

Term 1: c1 * x^p1 * y^q1

Term 2: c2 * x^p2 * y^q2

Term 3: c3 * x^p3 * y^q3

Linear and Constant Terms: c4*x + c5*y + c6

Evaluation Point (x, y)

What is the fxx fyy fxy fyx Calculator?

The fxx fyy fxy fyx Calculator is a tool designed to compute the second-order partial derivatives of a two-variable function, f(x, y), at a specific point. Specifically, it calculates:

• fxx: The second partial derivative of f with respect to x (∂²f/∂x²).
• fyy: The second partial derivative of f with respect to y (∂²f/∂y²).
• fxy: The mixed partial derivative of f, first with respect to x, then y (∂²f/∂y∂x).
• fyx: The mixed partial derivative of f, first with respect to y, then x (∂²f/∂x∂y).

This calculator is particularly useful for students of multivariable calculus, engineers, physicists, economists, and anyone dealing with functions of two variables where the rate of change of the rate of change is important. It helps in analyzing the concavity of a function, finding local extrema (using the second derivative test), and understanding the behavior of surfaces.

Our fxx fyy fxy fyx Calculator focuses on polynomial-like functions of the form f(x,y) = c1*x^p1*y^q1 + c2*x^p2*y^q2 + c3*x^p3*y^q3 + c4*x + c5*y + c6.

Who should use it?

Students learning multivariable calculus, teachers preparing examples, engineers analyzing system responses, and researchers modeling phenomena with two-variable functions will find this fxx fyy fxy fyx Calculator valuable.

Common Misconceptions

A common misconception is that fxy and fyx are always different. However, Clairaut’s Theorem states that if the second-order partial derivatives fxy and fyx are continuous in a region, then they are equal (fxy = fyx) in that region. Our fxx fyy fxy fyx Calculator assumes continuity and calculates fxy = fyx.

fxx fyy fxy fyx Formula and Mathematical Explanation

Given a function f(x, y), the first-order partial derivatives are fx (or ∂f/∂x) and fy (or ∂f/∂y). The second-order partial derivatives are obtained by differentiating the first-order partial derivatives:

• fxx = ∂/∂x (fx) = ∂²f/∂x²: Differentiate fx with respect to x.
• fyy = ∂/∂y (fy) = ∂²f/∂y²: Differentiate fy with respect to y.
• fxy = ∂/∂y (fx) = ∂²f/∂y∂x: Differentiate fx with respect to y.
• fyx = ∂/∂x (fy) = ∂²f/∂x∂y: Differentiate fy with respect to x.

For a term of the form c * x^p * y^q:

fx = c * p * x^p-1 * y^q

fy = c * q * x^p * y^q-1

Then:

fxx = c * p * (p-1) * x^p-2 * y^q

fyy = c * q * (q-1) * x^p * y^q-2

fxy = c * p * q * x^p-1 * y^q-1

fyx = c * q * p * x^p-1 * y^q-1 = fxy

The fxx fyy fxy fyx Calculator applies these rules to each term of the function f(x,y) = c1*x^p1*y^q1 + c2*x^p2*y^q2 + c3*x^p3*y^q3 + c4*x + c5*y + c6 and sums the results to find the total fxx, fyy, fxy, and fyx at the specified point (x,y).

Variables Table

Variable	Meaning	Unit	Typical Range
c1, c2, c3, c4, c5, c6	Coefficients of the terms	Dimensionless (depends on f)	Real numbers
p1, q1, p2, q2, p3, q3	Powers of x and y	Dimensionless	Real numbers (often non-negative integers)
x, y	Independent variables	Depends on context	Real numbers
fxx, fyy, fxy, fyx	Second-order partial derivatives	Units of f / (units of x/y)²	Real numbers

Practical Examples (Real-World Use Cases)

Example 1: Analyzing Concavity

Let f(x, y) = x³ + y³ – 3xy. We want to analyze its behavior around (1, 1).
So, c1=1, p1=3, q1=0; c2=1, p2=0, q2=3; c3=-3, p3=1, q3=1; c4=c5=c6=0.
fx = 3x² – 3y, fy = 3y² – 3x
fxx = 6x, fyy = 6y, fxy = -3, fyx = -3.
At (1, 1): fxx(1,1) = 6, fyy(1,1) = 6, fxy(1,1) = -3.
Using the calculator with c1=1, p1=3, q1=0; c2=1, p2=0, q2=3; c3=-3, p3=1, q3=1 and x=1, y=1, it will output fxx=6, fyy=6, fxy=-3.
The Hessian determinant D = fxx*fyy – (fxy)² = 6*6 – (-3)² = 36 – 9 = 27 > 0. Since D>0 and fxx>0, f(1,1) is a local minimum.

Example 2: Physics – Wave Equation

The wave equation involves second derivatives with respect to space (x) and time (t): ∂²u/∂t² = c² * ∂²u/∂x². While our calculator is for f(x,y), understanding second derivatives is crucial. If we had a function u(x,t), uxx and utt would be key.

Let’s take a function f(x,y) = sin(x)cos(y). This isn’t a polynomial, so our current calculator form doesn’t directly take sin/cos. However, if we approximated sin(x) ~ x and cos(y) ~ 1 – y²/2 near (0,0), f(x,y) ~ x(1-y²/2) = x – xy²/2.
So, c1=1, p1=1, q1=0; c2=-0.5, p2=1, q2=2. x=0, y=0.
fx = 1 – y²/2, fy = -xy
fxx = 0, fyy = -x, fxy = -y, fyx = -y
At (0,0): fxx=0, fyy=0, fxy=0, fyx=0.

How to Use This fxx fyy fxy fyx Calculator

1. Identify your function: Ensure your function f(x,y) can be represented in the form f(x,y) = c1*x^p1*y^q1 + c2*x^p2*y^q2 + c3*x^p3*y^q3 + c4*x + c5*y + c6.
2. Enter Coefficients and Powers: Input the values for c1, p1, q1 for the first term, c2, p2, q2 for the second, and so on, including c4, c5, and c6. If a term is missing, set its coefficient to 0.
3. Enter Evaluation Point: Input the x and y coordinates of the point at which you want to evaluate the derivatives.
4. Calculate: The calculator automatically updates as you type, or you can click “Calculate”.
5. Read Results: The values of fxx, fyy, fxy, and fyx at the specified point (x,y) are displayed, along with a bar chart and summary table.
6. Reset: Click “Reset” to return to default values.
7. Copy: Click “Copy Results” to copy the main outputs to your clipboard.

Decision-making Guidance

The signs and magnitudes of fxx, fyy, and fxy are crucial in the second derivative test for local extrema of f(x,y). The Hessian matrix uses these values to determine if a critical point is a local max, min, or saddle point. Our fxx fyy fxy fyx Calculator provides these values directly.

Key Factors That Affect fxx fyy fxy fyx Results

1. Function Form: The most significant factor is the function f(x,y) itself – the coefficients and powers directly dictate the form of the derivatives.
2. Point of Evaluation (x, y): The values of fxx, fyy, fxy, and fyx generally depend on the specific x and y values chosen.
3. Powers (p, q): Higher powers lead to more complex derivatives, and the values p-2, q-2 appearing in fxx, fyy mean the behavior can change drastically if p or q are less than 2.
4. Coefficients (c): These scale the magnitude of the derivatives.
5. Interaction between x and y terms: Mixed terms (where both x and y appear, like x^py^q with p, q > 0) are responsible for non-zero fxy and fyx.
6. Continuity: For fxy to equal fyx, the second partial derivatives need to be continuous. Polynomials are continuous everywhere, so this is satisfied for the functions used by this fxx fyy fxy fyx Calculator.

Frequently Asked Questions (FAQ)

Q1: What are second-order partial derivatives?

A1: They are derivatives of the first-order partial derivatives of a multivariable function, indicating the rate of change of the slope of the function along a particular axis or with respect to mixed variables.

Q2: Why are fxy and fyx often equal?

A2: Clairaut’s Theorem states that if fxy and fyx are continuous, they are equal. Polynomials and many common functions have continuous second partial derivatives.

Q3: What does fxx tell me about the function?

A3: fxx describes the concavity of the function f(x,y) in the x-direction (keeping y constant). A positive fxx suggests concave up, negative fxx suggests concave down.

Q4: What is the Hessian matrix?

A4: It’s a square matrix of second-order partial derivatives: [[fxx, fxy], [fyx, fyy]]. Its determinant is used in the second derivative test for local extrema.

Q5: Can this fxx fyy fxy fyx Calculator handle functions like sin(x) or e^y?

A5: No, this specific calculator is designed for polynomial-like terms of the form c*x^p*y^q, linear terms, and constants. You would need a more advanced symbolic differentiator for trigonometric or exponential functions, or use Taylor series approximations to fit the polynomial form near a point.

Q6: What if my function has more than three x^py^q terms?

A6: This calculator is limited to three such terms plus linear and constant parts. For more terms, you would need to calculate manually or use more advanced software.

Q7: How do I interpret the results for optimization?

A7: At a critical point (where fx=0 and fy=0), calculate D = fxx*fyy – (fxy)². If D>0 and fxx>0, local min. If D>0 and fxx<0, local max. If D<0, saddle point. If D=0, test is inconclusive.

Q8: What happens if p or q are less than 2?

A8: If p<2, x^p-2 will have a negative power, meaning x appears in the denominator for fxx. If p=0 or 1, the term vanishes after differentiating twice with respect to x. The calculator handles integer powers correctly.

Related Tools and Internal Resources

Explore more tools related to calculus and function analysis:

• Derivative Calculator: Find the derivative of single-variable functions.
• Integral Calculator: Compute definite and indefinite integrals.
• Polynomial Calculator: Work with polynomial functions, find roots, etc.
• Function Evaluator: Evaluate functions at given points.
• Limits Calculator: Find the limit of a function.
• Partial Derivative Calculator: Find first-order partial derivatives (a good precursor to using the fxx fyy fxy fyx Calculator).