Derivative of Multivariable Function

The derivative of a multivariable function is represented as a gradient vector, which contains all the partial derivatives of the function with respect to each of its input variables. This type of derivative indicates how the function changes along each axis in its domain and is differentiable within its domain. Key concepts include: - Partial Derivatives: These are derivatives of a function with respect to one variable while keeping other variables constant. - Gradient Vector: A vector that contains all the partial derivatives of the function. - Directional Derivative: Derivative of the function in a specific direction in the input space.

The derivative of a multivariable function f(x, y, ...) can be expressed as the gradient vector: ∇f = (∂f/∂x, ∂f/∂y, ...) This formula applies to functions with continuous partial derivatives within their domain.

We can derive the derivative of a multivariable function using different approaches. The fundamental methods include: - By First Principles: Involves taking limits of difference quotients for each partial derivative. - Using Chain Rule: Applies when functions are composed of other functions. - Using Total Differentiation: Combines all partial derivatives to understand changes in the function. By First Principles The derivative of a multivariable function can be shown using first principles by taking the limit of the difference quotient for each variable. Consider a function f(x, y). The partial derivative of f with respect to x is: ∂f/∂x = limΔx→0 [f(x + Δx, y) - f(x, y)] / Δx Similarly, the partial derivative with respect to y is: ∂f/∂y = limΔy→0 [f(x, y + Δy) - f(x, y)] / Δy Using Chain Rule For functions of the form f(g(x, y), h(x, y)), the chain rule is used to find derivatives with respect to x and y. - ∂f/∂x = (∂f/∂u)(∂u/∂x) + (∂f/∂v)(∂v/∂x) - ∂f/∂y = (∂f/∂u)(∂u/∂y) + (∂f/∂v)(∂v/∂y) Using Total Differentiation Total differentiation involves summing the effects of changes in each variable: df = (∂f/∂x)dx + (∂f/∂y)dy

Higher-order derivatives involve taking derivatives of the partial derivatives. These derivatives help analyze complicated changes in multivariable functions. Consider a function f(x, y): - The first derivative is ∇f = (∂f/∂x, ∂f/∂y). - The second-order derivatives include ∂²f/∂x², ∂²f/∂y², and mixed partial derivatives like ∂²f/∂x∂y. - Higher-order derivatives continue this pattern, offering insight into the curvature and behavior of the function.

- At points where the function is not differentiable, partial derivatives may not exist. - For functions with discontinuities, derivatives are undefined at those points.

- Partial Derivative: Derivative of a function with respect to one variable, holding others constant. - Gradient: A vector of partial derivatives indicating the direction of greatest increase of the function. - Directional Derivative: Rate of change of a function in a specific direction in its domain. - Mixed Partial Derivative: Derivative of a function with respect to two different variables. - Total Differentiation: Process of considering all partial derivatives to determine the overall rate of change of a function.