x1 x2 Derivative Formula

The derivative of a function represents the rate of change of the function’s output with respect to one of its inputs. When dealing with functions of multiple variables, such as x1 and x2, we often need to compute partial derivatives. The partial derivative of a function f with respect to x1, denoted as ∂f/∂x1, represents the rate of change of the function when x1 changes and all other variables are held constant. Similarly, the partial derivative with respect to x2, ∂f/∂x2, represents the rate of change when x2 changes and all other variables are held constant.

For a function f(x1, x2) of two variables, the partial derivatives can be computed using the following formulas:

∂f/∂x1 = ∂/∂x1 [f(x1, x2)] ∂f/∂x2 = ∂/∂x2 [f(x1, x2)]

To compute these partial derivatives, we treat the other variable as a constant. For example, when computing ∂f/∂x1, we treat x2 as a constant and differentiate the function with respect to x1 as we normally would.

Let’s consider a simple example to illustrate this concept. Suppose we have the function f(x1, x2) = 3x1^2 + 2x1x2 - 4x2^2. To find the partial derivatives, we would compute:

∂f/∂x1 = ∂/∂x1 (3x1^2 + 2x1x2 - 4x2^2) = 6x1 + 2x2

Here, we treated x2 as a constant and differentiated the function with respect to x1. The term -4x2^2 is treated as a constant because it does not depend on x1, and its derivative with respect to x1 is zero.

Similarly, to find the partial derivative with respect to x2, we would compute:

∂f/∂x2 = ∂/∂x2 (3x1^2 + 2x1x2 - 4x2^2) = 2x1 - 8x2

In this case, we treated x1 as a constant and differentiated the function with respect to x2.

These partial derivatives can be used to study the behavior of the function near a point, optimize functions of multiple variables, and model a wide range of phenomena in physics, engineering, economics, and other fields.

Second-Order Partial Derivatives

In addition to first-order partial derivatives, we can also compute second-order partial derivatives. These represent the rate of change of the first-order partial derivatives. For a function f(x1, x2), the four possible second-order partial derivatives are:

• ∂²f/∂x1² = ∂/∂x1 (∂f/∂x1)
• ∂²f/∂x2² = ∂/∂x2 (∂f/∂x2)
• ∂²f/∂x1∂x2 = ∂/∂x1 (∂f/∂x2)
• ∂²f/∂x2∂x1 = ∂/∂x2 (∂f/∂x1)

These second-order partial derivatives can provide additional information about the function, such as its concavity and the rate of change of its slope.

Example of Second-Order Partial Derivatives

Let’s return to the example function f(x1, x2) = 3x1^2 + 2x1x2 - 4x2^2. We previously computed the first-order partial derivatives:

∂f/∂x1 = 6x1 + 2x2 ∂f/∂x2 = 2x1 - 8x2

To find the second-order partial derivatives, we would compute:

∂²f/∂x1² = ∂/∂x1 (6x1 + 2x2) = 6 ∂²f/∂x2² = ∂/∂x2 (2x1 - 8x2) = -8 ∂²f/∂x1∂x2 = ∂/∂x1 (2x1 - 8x2) = 2 ∂²f/∂x2∂x1 = ∂/∂x2 (6x1 + 2x2) = 2

Note that ∂²f/∂x1∂x2 = ∂²f/∂x2∂x1, which is a result of Clairaut’s theorem. This theorem states that for functions with continuous second-order partial derivatives, the order of differentiation does not matter.

Conclusion

In conclusion, partial derivatives are a fundamental concept in multivariable calculus, allowing us to study the behavior of functions with multiple inputs. By computing partial derivatives, we can gain insights into the function’s rate of change, optimize functions, and model complex phenomena. The formulas and examples presented in this article demonstrate how to compute partial derivatives and apply them to real-world problems.

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