A partial derivative is a mathematical concept used to calculate the rate of change of a function with respect to one of its variables, while holding all other variables constant.
A partial derivative is denoted by the symbol ∂ (pronounced "partial") followed by the variable with respect to which the derivative is being taken. For example, the partial derivative of a function f(x,y) with respect to x would be written as ∂f/∂x.
A partial derivative only considers the rate of change of a function with respect to one variable, while a total derivative takes into account the effects of all variables on the function's rate of change. In other words, a total derivative is the sum of all partial derivatives for a function with multiple variables.
The chain rule for partial derivatives states that to find the partial derivative of a composite function, you must multiply the partial derivatives of each individual function in the chain. In other words, the partial derivative of f(g(x)) with respect to x is equal to the partial derivative of f with respect to g, multiplied by the partial derivative of g with respect to x.
Partial derivatives are used in many fields of science and engineering, including physics, economics, and engineering. They are particularly useful in calculating rates of change in complex systems, such as in climate models and financial forecasting.