Find Second Order Partial Derivatives Calculator

Calculate Second Order Partial Derivatives

For a function f(x, y) = ax² + bxy + cy² + dx + ey + f, find f_xx, f_yy, and f_xy.

Magnitudes of Second Order Partial Derivatives (|f_xx|, |f_yy|, |f_xy|)

What is a Second Order Partial Derivatives Calculator?

A second order partial derivatives calculator is a tool used to find the partial derivatives of a function with respect to its variables twice. For a function of two variables, say f(x, y), there are four second-order partial derivatives: f_xx (or ∂²f/∂x²), f_yy (or ∂²f/∂y²), f_xy (or ∂²f/∂x∂y), and f_yx (or ∂²f/∂y∂x). This particular calculator focuses on functions of the form f(x, y) = ax² + bxy + cy² + dx + ey + f, where a, b, c, d, e, and f are constants.

This type of calculator is invaluable for students learning multivariable calculus, engineers, physicists, and economists who deal with functions of multiple variables and need to analyze their rates of change and concavity or identify local extrema and saddle points.

Common misconceptions include thinking that f_xy and f_yx are always different. For most well-behaved functions (those with continuous second partial derivatives, like the polynomial we are using), Clairaut’s theorem states that f_xy = f_yx.

Second Order Partial Derivatives Formula and Mathematical Explanation

Given a function f(x, y), the first-order partial derivatives are:

• ∂f/∂x (or f_x): The derivative of f with respect to x, treating y as a constant.
• ∂f/∂y (or f_y): The derivative of f with respect to y, treating x as a constant.

The second-order partial derivatives are found by differentiating the first-order partial derivatives:

• ∂²f/∂x² (or f_xx): Differentiate f_x with respect to x.
• ∂²f/∂y² (or f_yy): Differentiate f_y with respect to y.
• ∂²f/∂x∂y (or f_xy): Differentiate f_y with respect to x (or f_x with respect to y, f_yx).

For our specific function f(x, y) = ax² + bxy + cy² + dx + ey + f:

1. First partial with respect to x (f_x):
   f_x = ∂/∂x (ax² + bxy + cy² + dx + ey + f) = 2ax + by + 0 + d + 0 + 0 = 2ax + by + d
2. First partial with respect to y (f_y):
   f_y = ∂/∂y (ax² + bxy + cy² + dx + ey + f) = 0 + bx + 2cy + 0 + e + 0 = bx + 2cy + e
3. Second partial f_xx:
   f_xx = ∂/∂x (f_x) = ∂/∂x (2ax + by + d) = 2a + 0 + 0 = 2a
4. Second partial f_yy:
   f_yy = ∂/∂y (f_y) = ∂/∂y (bx + 2cy + e) = 0 + 2c + 0 = 2c
5. Mixed partial f_xy:
   f_xy = ∂/∂y (f_x) = ∂/∂y (2ax + by + d) = 0 + b + 0 = b
6. Mixed partial f_yx:
   f_yx = ∂/∂x (f_y) = ∂/∂x (bx + 2cy + e) = b + 0 + 0 = b

As expected, f_xy = f_yx = b.

Variables Table

Variables used in the second order partial derivatives calculations.

Variable	Meaning	Unit	Typical Range
a, b, c, d, e, f	Coefficients of the polynomial f(x,y)	Dimensionless (or depends on context of f)	Any real number
fx, fy	First order partial derivatives	Units of f / units of x or y	Depends on x, y and coefficients
fxx, fyy, fxy, fyx	Second order partial derivatives	Units of f / (units of x or y)²	Depends on coefficients

Practical Examples (Real-World Use Cases)

Example 1: Analyzing a Cost Function

Suppose a company’s cost function is C(x, y) = 5x² + 3xy + 2y² + 50x + 30y + 1000, where x is units of product A and y is units of product B.

Here, a=5, b=3, c=2, d=50, e=30, f=1000.

Using the second order partial derivatives calculator (or the formulas):

• C_x = 10x + 3y + 50
• C_y = 3x + 4y + 30
• C_xx = 10
• C_yy = 4
• C_xy = 3

C_xx = 10 > 0 and C_yy = 4 > 0 suggest increasing marginal cost with respect to x and y individually. The Hessian matrix can be used with these values to find local minima or maxima for the cost.

Example 2: Shape of a Surface

Consider the surface defined by z = f(x, y) = -x² – y² + 2x + 4y – 3. Here, a=-1, b=0, c=-1, d=2, e=4, f=-3.

The second order partial derivatives calculator gives:

• f_x = -2x + 2
• f_y = -2y + 4
• f_xx = -2
• f_yy = -2
• f_xy = 0

f_xx = -2 and f_yy = -2 are both negative, indicating concavity downwards. Since f_xxf_yy – (f_xy)² = (-2)(-2) – 0² = 4 > 0, and f_xx < 0, there is a local maximum.

How to Use This Second Order Partial Derivatives Calculator

1. Identify the Coefficients: Look at your function f(x, y) and identify the values of a, b, c, d, e, and f corresponding to the terms ax², bxy, cy², dx, ey, and the constant f.
2. Enter Coefficients: Input these values into the respective fields in the calculator.
3. View Results: The calculator will automatically update and display the function f(x,y) based on your inputs, the first order partial derivatives (f_x and f_y), and the second order partial derivatives (f_xx, f_yy, and f_xy=f_yx).
4. Interpret Results: The values of f_xx, f_yy, and f_xy tell you about the concavity and the nature of critical points of the function. f_xx > 0 means concave up in the x-direction, f_xx < 0 means concave down.
5. Use the Chart: The bar chart visually represents the magnitudes of the second-order partial derivatives, helping you compare their relative influence.

Key Factors That Affect Second Order Partial Derivatives Results

The results of the second order partial derivatives calculator for the given polynomial form depend solely on the coefficients:

• Coefficient ‘a’: Directly determines f_xx = 2a. A larger ‘a’ means a larger second derivative with respect to x, indicating more pronounced curvature in the x-direction.
• Coefficient ‘c’: Directly determines f_yy = 2c. A larger ‘c’ means a larger second derivative with respect to y, indicating more pronounced curvature in the y-direction.
• Coefficient ‘b’: Directly determines f_xy = b. This mixed derivative relates to the twist or saddle nature of the surface.
• Coefficients ‘d’ and ‘e’: These affect the first-order partial derivatives but not the second-order ones for this specific polynomial form. They shift the location of critical points.
• Constant ‘f’: This shifts the function up or down but does not affect any of the partial derivatives.
• Function Form: This calculator is specifically for f(x, y) = ax² + bxy + cy² + dx + ey + f. More complex functions will have second-order partial derivatives that also depend on x and y, not just constants.

Frequently Asked Questions (FAQ)

What are second order partial derivatives?

They are the derivatives of the first order partial derivatives of a multivariable function, indicating the rate of change of the slope of the function in specific directions and its concavity.

Why are f_xy and f_yx usually equal?

According to Clairaut’s Theorem (or Young’s Theorem), if the second partial derivatives are continuous in a region, then the order of differentiation does not matter, so f_xy = f_yx.

What do f_xx and f_yy tell us?

f_xx tells us about the concavity of the function along the x-direction (holding y constant), and f_yy tells us about the concavity along the y-direction (holding x constant). Positive values suggest concavity upwards (like a minimum), negative suggest concavity downwards (like a maximum).

How are second order partial derivatives used to find local extrema?

The Second Derivative Test for multivariable functions uses f_xx, f_yy, and f_xy (in the form of the Hessian determinant D = f_xxf_yy – (f_xy)²) at a critical point to classify it as a local minimum, maximum, or saddle point. Check our local extrema calculator.

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

No, this specific second order partial derivatives calculator is designed for polynomial functions of the form f(x, y) = ax² + bxy + cy² + dx + ey + f. For other functions, the derivatives would be different and might depend on x and y.

What if my function has higher order terms like x³ or y³?

This calculator won’t directly work. You would need a more advanced symbolic differentiator or perform the differentiation manually based on the rules of partial differentiation.

What does a zero second partial derivative mean?

If f_xx=0 at a point, it might indicate an inflection point in the x-direction, but you need to look at higher-order derivatives or other information.

Where can I learn more about partial derivatives?

You can explore resources on multivariable calculus, which covers partial derivatives in detail.

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