A partial derivative is a type of derivative that measures the rate of change of a function with respect to one of its variables, while holding all other variables constant. It is useful in multi-variable calculus, where functions have more than one independent variable.
The notation for partial derivatives is similar to that of regular derivatives, but with a subscript denoting which variable is being held constant. For example, the partial derivative of a function f(x,y) with respect to x is written as ∂f/∂x, and the partial derivative with respect to y is written as ∂f/∂y.
To find the partial derivative of a function, you take the derivative of the function with respect to the variable in question, treating all other variables as constants. This can be done using the standard rules of differentiation, such as the power rule and the product rule.
The chain rule also applies to partial derivatives. This means that when a function is composed of multiple sub-functions, the partial derivative of the composite function can be found by taking the partial derivatives of each sub-function and multiplying them together.
The given equation, bz(x)=az(y), represents a function that is dependent on two variables, x and y. To show this using partial derivatives, we can take the partial derivative of bz(x) with respect to x, which would be b, and the partial derivative of bz(x) with respect to y, which would be 0. Similarly, taking the partial derivative of az(y) with respect to x would result in 0, and the partial derivative with respect to y would be a. Therefore, the two sides of the equation are equal, showing that bz(x) and az(y) are both partial derivatives of the same function.