Calculate all four second-order partial derivatives and...

Calculate all four second-order partial derivatives and check that $f_{x y}=f_{y x} .$ Assume the variables are restricted to a domain on which the function is defined.
$$f(x, y)=3 \sin 2 x \cos 5 y$$

Key Concepts

Partial Derivatives

Partial derivatives refer to the derivatives of a function with respect to one variable while holding the other variables constant. They measure the rate of change of the function in the direction of one coordinate axis and are foundational to multivariable calculus.

Mixed Partial Derivatives

Mixed partial derivatives are the derivatives obtained by differentiating with respect to two different variables in sequence. For example, differentiating first with respect to x and then y (and vice versa) gives f_xy and f_yx. These derivatives help in understanding how the rate of change in one direction influences the rate of change in another direction.

Clairaut's Theorem

Clairaut's Theorem (also known as the equality of mixed partials) states that if the mixed partial derivatives are continuous on a domain, then the order of differentiation does not matter, i.e., f_xy will equal f_yx. This theorem is essential to verifying the consistency of computations involving second-order derivatives in many applications.

Chain Rule

The chain rule is a fundamental differentiation technique used when dealing with composite functions. In the context of finding partial derivatives of functions like f(x, y) where variables are inside other functions (e.g., trigonometric functions with linear functions of x and y), the chain rule facilitates the calculation by taking into account the derivative of the inner function.

Second-Order Partial Derivatives

Second-order partial derivatives involve taking the derivative of a first-order partial derivative. These derivatives can be taken twice with respect to the same variable, or with respect to two different variables, providing important information about the curvature and the behavior of the function in the neighborhood of a point.

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