Understanding Partial Derivatives of e^(x^2y^2)

When dealing with multivariable functions, partial derivatives play a crucial role in understanding how the function changes with respect to each variable while keeping others constant. In this article, we will explore the partial derivatives of the function e^(x^2y^2) with respect to both x and y.

Introduction to Partial Derivatives

Partial derivatives of a function are the derivatives of the function with respect to one of its variables while all other variables are held constant. This allows us to analyze the function's behavior along different dimensions.

Chain Rule and the Derivative of e^(f(x))

The chain rule is a fundamental concept in calculus that helps us find the derivative of a composite function. According to the chain rule, if we have a function e^(f(x)), the derivative of this function with respect to x is given by:

(frac{d}{dx} e^{f(x)} e^{f(x)} cdot f'(x))

Partial Derivative with Respect to x

To find the partial derivative of e^(x^2 y^2) with respect to x, we treat y as a constant. Using the chain rule, we can break this down as follows:

Let f(x, y) x^2 y^2.

The derivative of x^2 y^2 with respect to x is:

(frac{partial}{partial x} (x^2 y^2) 2x y^2)

Thus, applying the chain rule:

(frac{partial}{partial x} e^{x^2 y^2} e^{x^2 y^2} cdot 2x y^2)

This can be simplified to:

(frac{partial}{partial x} e^{x^2 y^2} 2x y^2 e^{x^2 y^2})

Partial Derivative with Respect to y

Similarly, to find the partial derivative of e^(x^2 y^2) with respect to y, we treat x as a constant. Again, using the chain rule:

The derivative of x^2 y^2 with respect to y is:

(frac{partial}{partial y} (x^2 y^2) 2x^2 y)

Applying the chain rule:

(frac{partial}{partial y} e^{x^2 y^2} e^{x^2 y^2} cdot 2x^2 y)

This can be simplified to:

(frac{partial}{partial y} e^{x^2 y^2} 2x^2 y e^{x^2 y^2})

Conclusion

Understanding the partial derivatives of the function e^(x^2 y^2) is essential for analyzing the behavior of such complex functions. By applying the chain rule, we can derive the specific partial derivatives, which are:

(frac{partial}{partial x} e^{x^2 y^2} 2x y^2 e^{x^2 y^2}) (frac{partial}{partial y} e^{x^2 y^2} 2x^2 y e^{x^2 y^2})

These derivatives provide valuable insights into the function's behavior along different dimensions and can be applied in various fields, such as physics, engineering, and economics.