Find The Four Second Partial Derivatives Of The Following Function

Introduction

Finding the four second partial derivatives of the following function f(x, y) = x² y + 3 x y² is a fundamental exercise in multivariable calculus. These derivatives—(f_{xx}), (f_{yy}), (f_{xy}), and (f_{yx})—describe how the rate of change of a function varies with respect to each variable and their combinations. Mastering this process builds a solid foundation for optimization, physics, economics, and engineering applications That's the part that actually makes a difference. No workaround needed..

Quick note before moving on.

Steps

Step 1: Compute the first‑order partial derivatives

1. Partial derivative with respect to x
   [ f_x = \frac{\partial}{\partial x}\bigl(x^2 y + 3 x y^2\bigr) = 2x y + 3 y^2 ]
2. Partial derivative with respect to y
   [ f_y = \frac{\partial}{\partial y}\bigl(x^2 y + 3 x y^2\bigr) = x^2 + 6 x y ]

These first‑order results are the starting point for any second‑order analysis.

Step 2: Differentiate the first‑order results again

1. Second derivative with respect to x (fₓₓ)
   [ f_{xx} = \frac{\partial}{\partial x}\bigl(2x y + 3 y^2\bigr) = 2y ]
2. Second derivative with respect to y (fᵧᵧ)
   [ f_{yy} = \frac{\partial}{\partial y}\bigl(x^2 + 6 x y\bigr) = 6x ]
3. Mixed derivative fₓᵧ (differentiate (f_x) with respect to y)
   [ f_{xy} = \frac{\partial}{\partial y}\bigl(2x y + 3 y^2\bigr) = 2x + 6y ]
4. Mixed derivative fᵧₓ (differentiate (f_y) with respect to x)
   [ f_{yx} = \frac{\partial}{\partial x}\bigl(x^2 + 6 x y\bigr) = 2x + 6y ]

Notice that (f_{xy} = f_{yx}); this equality is guaranteed by Clairaut’s theorem when the function is continuous and its partial derivatives are continuous.

Step 3: Verify continuity conditions

For polynomial functions like the one above, all first‑order partial derivatives are polynomials, hence continuous everywhere. Which means, the equality of mixed partials holds universally for this example.

Step 4: Summarize the four second partial derivatives

• (f_{xx}) = 2 y
• (f_{yy}) = 6 x
• (f_{xy}) = 2 x + 6 y
• (f_{yx}) = 2 x + 6 y

These expressions fully answer the request to find the four second partial derivatives of the following function.

Scientific Explanation

Second partial derivatives provide insight into the curvature of a surface defined by a multivariable function Which is the point..

• (f_{xx}) measures the rate at which the slope in the x direction changes as you move along x, holding y constant. A positive value indicates the surface bends upward (convex) in that direction, while a negative value signals a downward bend (concave).
• (f_{yy}) performs the analogous role for the y direction.
• Mixed partials ((f_{xy}) and (f_{yx})) capture how the slope in one

direction influences the slope in the perpendicular direction. This cross-term reveals how changes in x can tilt or twist the surface in the y direction, and vice versa. Together, these four second-order derivatives form the Hessian matrix, a critical tool for classifying critical points (maxima, minima, saddle points) in multivariable optimization.

In physics, second partial derivatives appear in partial differential equations such as the wave equation and Laplace’s equation, where they describe how quantities like displacement, temperature, or electric potential change spatially. In economics, they help model marginal effects and curvature in utility or production functions, guiding decisions about resource allocation. Engineers use them to analyze stress tensors and material deformation, where directional sensitivity determines structural integrity.

Conclusion

Computing second partial derivatives is a foundational skill in multivariable calculus with far-reaching implications. The equality of mixed partials, guaranteed by Clairaut’s theorem under continuity, ensures consistency in these analyses. By systematically differentiating a function twice—first to find the slopes, then to analyze how those slopes evolve—we uncover detailed information about a surface’s shape and behavior. Whether optimizing design parameters, modeling natural phenomena, or solving complex systems, second-order derivatives provide the mathematical lens through which we understand curvature, interaction, and dynamic change in multidimensional spaces That alone is useful..