Find Partial Derivative Fxy Calculator

Find fxy for f(x, y) = A * x^a * y^b

This calculator finds the mixed second-order partial derivative f_xy for a function of the form f(x, y) = A * x^a * y^b. Enter the coefficient A, and exponents a and b.

f_xy Values at Different Points

x	y	fxy(x, y)
0.8	0.8	6.144
0.9	0.9	8.748
1	1	12
1.1	1.1	15.972
1.2	1.2	20.736

Table showing f_xy evaluated at various x and y values around the input point.

What is a Partial Derivative fxy Calculator?

A Partial Derivative fxy Calculator is a specialized tool designed to compute the mixed second-order partial derivative of a function of two variables, f(x, y), first with respect to x and then with respect to y (denoted as f_xy or ∂²f/∂y∂x). Our find partial derivative fxy calculator focuses on functions of the form f(x, y) = A * x^a * y^b, which are common in various mathematical and scientific fields.

Essentially, f_xy measures how the rate of change of f with respect to x (which is f_x) itself changes as y varies. It helps understand the interdependency of the rates of change along different axes.

Anyone studying multivariable calculus, physics, engineering, economics, or any field that models systems with multiple interacting variables should use this partial derivative fxy calculator. It’s particularly useful for students learning about partial differentiation and for professionals needing quick calculations for specific function forms. Common misconceptions include thinking f_xy is always equal to f_yx (it is, under certain continuity conditions – Clairaut’s Theorem), or that it’s the same as f_xx or f_yy.

Partial Derivative fxy Formula and Mathematical Explanation

For a function of two variables f(x, y), the first partial derivative with respect to x, f_x (or ∂f/∂x), is found by differentiating f with respect to x while treating y as a constant.

For our specific case, f(x, y) = A * x^a * y^b:

1. First partial derivative with respect to x (f_x):
f_x = ∂/∂x (A * x^a * y^b) = A * y^b * ∂/∂x (x^a) = A * y^b * (a * x^a-1) = A * a * x^a-1 * y^b

2. Second partial derivative f_xy (differentiating f_x with respect to y):
f_xy = ∂/∂y (f_x) = ∂/∂y (A * a * x^a-1 * y^b)
Now we treat x (and thus x^a-1) as constant:
f_xy = A * a * x^a-1 * ∂/∂y (y^b) = A * a * x^a-1 * (b * y^b-1) = A * a * b * x^a-1 * y^b-1

So, the formula used by the find partial derivative fxy calculator is f_xy = A * a * b * x^a-1 * y^b-1.

Variables in the Formula

Variable	Meaning	Unit	Typical Range
A	Coefficient of the 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
x, y	Independent variables	Depends on the context	Any real number where f is defined
fx	First partial derivative w.r.t. x	Units of f / Units of x	Varies
fxy	Mixed second partial derivative	Units of f / (Units of x * Units of y)	Varies

Practical Examples (Real-World Use Cases)

Let’s see how our partial derivative fxy calculator can be used.

Example 1: Production Function

Suppose a company’s production (P) is modeled by P(K, L) = 10 * K^0.5 * L^0.5, where K is capital and L is labor. Here, A=10, a=0.5, b=0.5.

Using the formula or the find partial derivative fxy calculator:

• P_K = 10 * 0.5 * K^-0.5 * L^0.5 = 5 * K^-0.5 * L^0.5
• P_KL = 5 * K^-0.5 * 0.5 * L^-0.5 = 2.5 * K^-0.5 * L^-0.5

If K=100 and L=400, P_KL = 2.5 * (100)^-0.5 * (400)^-0.5 = 2.5 * (1/10) * (1/20) = 2.5 / 200 = 0.0125. This positive value indicates that increasing labor increases the marginal product of capital (and vice-versa).

Example 2: Temperature Distribution

Imagine the temperature T on a metal plate is given by T(x, y) = 50 * x² * y, where x and y are coordinates. Here A=50, a=2, b=1.

Using the partial derivative fxy calculator logic:

• T_x = 50 * 2 * x¹ * y¹ = 100 * x * y
• T_xy = 100 * x * 1 * y⁰ = 100 * x

At the point (2, 3), T_xy = 100 * 2 = 200. This tells us how the rate of change of temperature along x changes as we move along y at that point.

How to Use This Partial Derivative fxy Calculator

Using our find partial derivative fxy calculator is straightforward:

1. Enter Coefficient (A): Input the multiplicative constant A of your term.
2. Enter Exponent of x (a): Input the power ‘a’ to which x is raised.
3. Enter Exponent of y (b): Input the power ‘b’ to which y is raised.
4. Enter Value of x (Optional): If you want to evaluate f_xy at a specific point, enter the x-coordinate.
5. Enter Value of y (Optional): Similarly, enter the y-coordinate for evaluation.
6. Calculate: Click “Calculate” or observe the real-time update.

The results will show:

• The expression for f_xy.
• The expression for f_x.
• The value of f_xy at the specified (x, y) point, if provided.

You can use the “Reset” button to return to default values and “Copy Results” to copy the output. The chart and table also update dynamically based on your inputs, especially the evaluation point (x, y). Explore our calculus resources for more tools.

Key Factors That Affect fxy Results

The value and form of f_xy are directly influenced by:

1. Coefficient A: This directly scales f_xy. A larger A means a larger magnitude for f_xy.
2. Exponent a (of x): If a=0 or a=1, the x-dependence in f_xy simplifies or disappears (as x^a-1 becomes x^-1 or x⁰). If ‘a’ is less than 1, f_xy might involve negative exponents of x.
3. Exponent b (of y): Similar to ‘a’, if b=0 or b=1, the y-dependence in f_xy simplifies or vanishes (as y^b-1 becomes y^-1 or y⁰).
4. Values of x and y: If evaluating f_xy at a point, the specific x and y values determine the numerical result, especially when a-1 or b-1 are non-zero.
5. Sign of A, a, b: The signs of these parameters determine the sign of f_xy.
6. Magnitude of a and b: Larger magnitudes of a and b generally lead to larger magnitudes of f_xy, especially when |x| and |y| are greater than 1. Learn more about differentiation rules.

Understanding these helps interpret the output of the partial derivative fxy calculator.

Frequently Asked Questions (FAQ)

What does f_xy represent intuitively?

It measures how the slope of the function in the x-direction changes as you move in the y-direction.

Is f_xy always equal to f_yx?

If f_xy and f_yx are continuous in a region, then yes, they are equal in that region (Clairaut’s Theorem on equality of mixed partials). Our find partial derivative fxy calculator deals with functions where this is true.

What if a or b is 0 or 1?

If a=0, f doesn’t depend on x initially, so fx and fxy are 0. If b=0, f doesn’t depend on y, so fxy is 0. If a=1, a-1=0, so x^0=1 in fxy. If b=1, b-1=0, y^0=1 in fxy.

Can this calculator handle functions like sin(xy) or e^(x+y)?

No, this specific partial derivative fxy calculator is designed ONLY for f(x, y) = A * x^a * y^b. Other functions require different differentiation rules.

What if a-1 or b-1 is negative?

The calculator handles this, resulting in x or y terms in the denominator of f_xy. For example, if a=0.5, a-1=-0.5, so x^-0.5 = 1/√x.

Can I use non-integer values for a and b?

Yes, the exponents ‘a’ and ‘b’ can be any real numbers you input into the find partial derivative fxy calculator.

What does a zero value for f_xy mean?

It means the rate of change of f with respect to x (f_x) does not change as y varies at that point, or f_x is independent of y.

Where are mixed partial derivatives used?

They are used in physics (e.g., wave equation, heat equation), economics (e.g., utility functions, production functions), optimization problems with multiple variables, and differential geometry. See our section on applications of calculus.

Related Tools and Internal Resources

• Calculus Basics: Learn fundamental concepts of differentiation and integration.
• Differentiation Rules Guide: A comprehensive guide to various rules of differentiation.
• Applications of Partial Derivatives: Explore where partial derivatives are used in real-world scenarios.
• First Derivative Calculator: Calculate the first derivative of single-variable functions.
• Second Derivative Calculator: Find f_xx or f_yy for single or multivariable functions (if available).
• Gradient Calculator: Calculate the gradient of a multivariable function.