Partial differentiation

Section: Understanding Partial Differentiation

  • Partial differentiation refers to the process of differentiating a function with respect to one variable, treating all others as constants.

  • Every term without the variable of interest is treated as a constant term, while all terms that do contain the variable are differentiated in the standard way.

  • The notation for a partial derivative uses a curly “d”, denoted as ∂f/∂x or ∂f/∂y. The denominator represents the variable with respect to which the differentiation is carried out.

  • An important concept in partial differentiation is the order of differentiation. A first order partial derivative differentiates the function with respect to one variable. A second order derivative differentiates the function twice.

Section: Applications of Partial Differentiation

  • Partial derivatives can be used to find tangent planes to surfaces at a specific point.

  • The gradient vector, which indicates the direction of steepest increase for a function at a given point, can be found by taking the partial derivatives with respect to all variables and forming a vector with these values.

  • Optimisation problems often involve finding the maximum or minimum of a function, which can be solved by setting the first order partial derivatives of the function equal to zero and solving for the variables.

  • In many physical sciences, the change of a variable is studied in relation to changes in other variables. For these analyses, partial differentiation tools are often indispensable.

Section: Higher Order Partial Differentiation

  • Higher order partial derivatives involves differentiating a function more than once, but with respect to different variables.

  • The notation for a second order mixed derivative (differentiating first with respect to x, then y) is denoted as ∂²f/∂x∂y or ∂²f/∂y∂x.

  • An important result is Clairoaut’s theorem, which states that as long as a function’s mixed second order partial derivatives are continuous, then the mixed second order partial derivatives of a function in relation to two variables is independent of the order of differentiation. It is often expressed as ∂²f/∂x∂y = ∂²f/∂y∂x.