Find The Second Order Partial Derivative Calculator

Calculate f_xx, f_yy, and f_xy for f(x,y) = Ax^ay^b + C at a specified point.

Calculator

Enter the function f(x,y) = Ax^ay^b + C and the point (x,y) for evaluation:

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Results

For the function f(x,y) = 1*x^2*y^1 + 0 and point (1, 1):

The first partial derivatives are f_x = A*a*x^a-1y^b and f_y = A*b*x^ay^b-1. The second partial derivatives are f_xx = A*a*(a-1)*x^a-2y^b, f_yy = A*b*(b-1)*x^ay^b-2, and f_xy = f_yx = A*a*b*x^a-1y^b-1.

Derivatives Summary

Derivative	Symbol	Formula (for Ax^ay^b+C)	Value at (1, 1)
First partial w.r.t x	fx	2xy	2
First partial w.r.t y	fy	x^2	1
Second partial w.r.t x, x	fxx	2y	2
Second partial w.r.t y, y	fyy	0	0
Second partial w.r.t x, y	fxy	2x	2
Second partial w.r.t y, x	fyx	2x	2

Table showing first and second partial derivatives and their values at the specified point.

What is a Second Order Partial Derivative?

A Second Order Partial Derivative measures the rate of change of a first partial derivative of a function with multiple variables. For a function f(x, y), the first partial derivatives f_x and f_y tell us how the function changes as x or y changes, respectively. The Second Order Partial Derivatives (f_xx, f_yy, f_xy, f_yx) tell us how these rates of change (f_x and f_y) are themselves changing.

For example, f_xx measures the rate of change of f_x as x changes (holding y constant), indicating the “curvature” of the function in the x-direction. f_xy measures the rate of change of f_x as y changes (holding x constant).

These derivatives are crucial in optimization problems (finding maxima and minima), in physics (like wave equations and heat equations), and in economics to understand the relationships between multiple variables.

Anyone studying multivariable calculus, physics, engineering, or economics will encounter and use the Second Order Partial Derivative.

A common misconception is that f_xy and f_yx are always different. However, for most well-behaved functions (those with continuous second partial derivatives), Clairaut’s Theorem states that f_xy = f_yx (the order of differentiation does not matter).

Second Order Partial Derivative Formula and Mathematical Explanation

For a function f(x, y), the first partial derivatives are:

• f_x = ∂f/∂x (differentiate f with respect to x, treating y as a constant)
• f_y = ∂f/∂y (differentiate f with respect to y, treating x as a constant)

The Second Order Partial Derivatives are found by differentiating the first partial derivatives:

• f_xx = ∂/∂x (∂f/∂x) = ∂²f/∂x² (differentiate f_x with respect to x)
• f_yy = ∂/∂y (∂f/∂y) = ∂²f/∂y² (differentiate f_y with respect to y)
• f_xy = ∂/∂y (∂f/∂x) = ∂²f/∂y∂x (differentiate f_x with respect to y)
• f_yx = ∂/∂x (∂f/∂y) = ∂²f/∂x∂y (differentiate f_y with respect to x)

For our calculator’s specific function form, f(x, y) = Ax^ay^b + C:

• f_x = A * a * x^a-1y^b
• f_y = A * b * x^ay^b-1
• f_xx = A * a * (a-1) * x^a-2y^b
• f_yy = A * b * (b-1) * x^ay^b-2
• f_xy = A * a * b * x^a-1y^b-1
• f_yx = A * b * a * x^ay^b-1 = f_xy

Variables Table

Variable	Meaning	Unit	Typical Range
A	Coefficient of the x^ay^b term	Depends on the context of f	Any real number
a	Exponent of x	Dimensionless	Any real number
b	Exponent of y	Dimensionless	Any real number
C	Constant term	Depends on the context of f	Any real number
x, y	Independent variables of f	Depends on the context of f	Any real numbers (where defined)

Variables used in the function f(x,y) = Ax^ay^b + C and its derivatives.

Practical Examples (Real-World Use Cases)

Example 1: Analyzing a Cost Function

Suppose a company’s production cost C(x, y) is given by C(x, y) = 50x²y + 200, where x is the amount of labor and y is the amount of raw material. We want to find the second-order partial derivatives at x=10, y=5.

Here, A=50, a=2, b=1, C=200, x=10, y=5.

C_x = 100xy, C_y = 50x²

C_xx = 100y, C_yy = 0, C_xy = 100x

At (10, 5): C_xx = 100*5 = 500, C_yy = 0, C_xy = 100*10 = 1000.

C_xx = 500 means the rate of change of marginal cost with respect to labor is increasing as labor increases (at y=5). C_xy=1000 indicates how the marginal cost with respect to labor changes as raw material increases.

Example 2: Temperature Distribution

Imagine the temperature T(x, y) on a metal plate is given by T(x, y) = 100 – 0.5x² – 1.5y². We are interested in the curvature of the temperature profile at (1, 1). This fits the form if we consider separate terms, but our calculator handles one term plus a constant. Let’s simplify and assume part of the temperature profile is like T(x,y) = -0.5x² + (-1.5y²), but our calculator uses x*y term. Let’s use T(x,y) = -0.5x²y⁰ – 1.5x⁰y² + 100. Our calculator is for Ax^ay^b+C. So, let’s use a function like T(x,y) = -2x¹y¹ + 100. A=-2, a=1, b=1, C=100. Point (2,3).

T_x = -2y, T_y = -2x

T_xx = 0, T_yy = 0, T_xy = -2

At (2, 3): T_xx = 0, T_yy = 0, T_xy = -2. The zero values for T_xx and T_yy come from the powers being 1. If we had T(x,y) = -2x²y³ + 100, then T_xx, T_yy would be non-zero. The Second Order Partial Derivative helps understand heat flow and distribution.

How to Use This Second Order Partial Derivative Calculator

1. Enter Function Parameters: Input the values for A, a, b, and C to define your function f(x,y) = Ax^ay^b + C.
2. Enter Evaluation Point: Input the x and y coordinates (x_val, y_val) at which you want to evaluate the derivatives.
3. View Results: The calculator automatically updates and displays the first partial derivatives (f_x, f_y) and the Second Order Partial Derivatives (f_xx, f_yy, f_xy, f_yx), both as functions and evaluated at your specified point.
4. Analyze Table and Chart: The table summarizes the derivatives, and the chart visualizes the function’s behavior near the point.
5. Reset or Copy: Use the “Reset” button to return to default values or “Copy Results” to copy the output.

The results tell you about the local behavior of the function f(x,y) around the point (x_val, y_val), such as its concavity and how its slope changes.

Key Factors That Affect Second Order Partial Derivative Results

1. The Function Form: The values of A, a, b, and C directly determine the expressions for the derivatives. Higher powers (a, b) lead to more complex derivative expressions.
2. The Values of Exponents (a, b): If ‘a’ or ‘b’ are less than 2, f_xx or f_yy might become zero or involve negative exponents. If ‘a’ or ‘b’ are less than 1, f_x, f_y or f_xy might involve negative exponents.
3. The Coefficient (A): This scales all the derivatives proportionally.
4. The Constant (C): The constant C disappears after the first differentiation, so it does not affect any partial derivatives.
5. The Point of Evaluation (x_val, y_val): The specific values of x and y at which the derivatives are evaluated determine their numerical results. If negative exponents appear and x or y is zero, the derivative might be undefined at that point.
6. Continuity of Derivatives: For f_xy to equal f_yx, the second partial derivatives need to be continuous, which is true for polynomial-like functions away from points where denominators become zero.

Frequently Asked Questions (FAQ)

Q: What does f_xx represent?

A: f_xx represents the rate of change of the slope in the x-direction as x changes. It indicates the concavity of the function along a line parallel to the x-axis passing through the point (x,y).

Q: What does f_xy represent?

A: f_xy represents how the slope in the x-direction (f_x) changes as y changes. It’s a “mixed” partial derivative.

Q: When is f_xy equal to f_yx?

A: According to Clairaut’s Theorem (or Schwarz’s theorem), if the second partial derivatives are continuous in a region around a point, then f_xy = f_yx at that point.

Q: What if the power ‘a’ or ‘b’ is less than 2?

A: If ‘a’ is 0 or 1, f_xx will be zero. If ‘a’ is between 0 and 2 (but not 0 or 1), f_xx will involve x raised to a negative power.

Q: Can I use this calculator for functions other than Ax^ay^b + C?

A: This specific calculator is designed for functions of the form Ax^ay^b + C. For more complex functions (like those involving sin, cos, exp, log, or sums of many terms), you would need a more advanced symbolic differentiator or apply the rules of differentiation manually to each term and then sum them up.

Q: What does a positive f_xx mean?

A: A positive f_xx at a point suggests the function is concave up (like a valley) in the x-direction around that point.

Q: What does a negative f_xx mean?

A: A negative f_xx at a point suggests the function is concave down (like a hill) in the x-direction around that point.

Q: How are Second Order Partial Derivatives used in optimization?

A: In finding local maxima or minima of f(x,y), we first find critical points where f_x=0 and f_y=0. Then, the Second Derivative Test uses f_xx, f_yy, and f_xy (the Hessian matrix) to classify these points as local max, min, or saddle points.