What is the Second Order Partial Derivative of w x2tan(xy)?

Introduction: This article delves into the second-order partial derivatives of the function w x2tan(xy). We will explore both the first-order and second-order partial derivatives and illustrate the process of finding them.

First Order Partial Derivatives

The first order partial derivatives of the function w x2tan(xy) can be found using the product rule and the chain rule from calculus.

First Order Partial Derivative with respect to x:

Using the product rule and the chain rule, we find:

wx x2ysec2(xy) 2x tan(xy)

First Order Partial Derivative with respect to y:

Again using the product and chain rules, we obtain:

wy x3sec2(xy)

Second Order Partial Derivatives

Next, we compute the second order partial derivatives of w, which are the partial derivatives of the first order partial derivatives.

Second Order Partial Derivative with respect to x (wxx):

When we take the second partial derivative with respect to x, we use the product and chain rules again:

wxx x2y2sec2(xy) - 2x2y tan2(xy) - 4xy sec2(xy) 2ytan(xy)

Second Order Partial Derivative with respect to y (wyy):

For the partial derivative with respect to y, the result is:

wyy x4sec2(xy) - 2x4tan2(xy)

Second Order Cross Partial Derivative (wxy and wyx):

The cross partial derivative for both x and y is the same:

wxy wyx x3y sec2(xy) - 3x2 sec2(xy)

Verification and Additional Insights

It's crucial to verify the correctness of the calculations, as detailed in the provided equations. The steps involved are complex and prone to errors without careful consideration. Hence, double-checking the results is recommended.

Additional Tips:

When working with such functions, it is often helpful to:

Identify the appropriate rule for differentiation (product rule, chain rule, etc.). Remember that for most well-behaved functions, mixed partial derivatives (wxy and wyx) are equal. Ensure all trigonometric identities are correctly applied.

Understanding these principles and the process of finding partial derivatives is essential in fields such as physics, engineering, and higher mathematics.

Conclusion

In conclusion, the second order partial derivatives of the function w x2tan(xy) involve applying higher-order differentiation techniques appropriately. Mastering these techniques can greatly enhance one's problem-solving skills in various scientific and mathematical contexts.