What are all the the notations for partial derivatives where z=f(x,y)?
The notations for partial derivatives where z=f(x,y) are:
- ∂z/∂x
- f_x
- D_1f
- ∂_1f
- ∂f/∂x
- ∂z/∂y
- f_y
- D_2f
- ∂_2f
- ∂f/∂y
There are several notations for partial derivatives. Let's assume z is a function of x and y, expressed as z = f(x, y). The most common notations for partial derivatives are:
1. ∂z/∂x: This notation represents the partial derivative of z with respect to x, keeping y constant. It indicates how z varies with changes in x, while treating y as a constant.
2. ∂z/∂y: This notation represents the partial derivative of z with respect to y, keeping x constant. It indicates how z varies with changes in y, while treating x as a constant.
3. ∂²z/∂x²: This notation represents the second partial derivative of z with respect to x. It indicates how the rate of change of z with respect to x changes with respect to x.
4. ∂²z/∂y²: This notation represents the second partial derivative of z with respect to y. It indicates how the rate of change of z with respect to y changes with respect to y.
5. ∂²z/∂x∂y or ∂²z/∂y∂x: These notations represent the mixed partial derivatives of z with respect to x and y. Both notations are used interchangeably and indicate how the rate of change of z with respect to x changes with respect to y.
6. Dz/Dx: This notation represents the total derivative of z with respect to x, meaning it considers the effect of both x and y on z and accounts for their combined variation.
Note that all these notations indicate partial derivatives, except for Dz/Dx, which represents the total derivative.
To denote partial derivatives in mathematics, there are a few different notations that can be used depending on personal preference or conventions followed in different contexts. I'll explain three commonly used notations for partial derivatives.
1. **Using Subscript Notation**: In this notation, partial derivatives are denoted by adding subscripts to the variable symbol. For example, if you have a function z = f(x, y), the partial derivative of z with respect to x can be written as ∂z/∂x or f*_x_*.
2. **Using Fraction Notation**: This notation represents partial derivatives using fractions. The numerator represents the derivative, and the denominator represents the variable with respect to which the derivative is being taken. Using this notation, the partial derivative of z with respect to x can be expressed as ∂z/∂x or df/dx.
3. **Using Leibniz Notation**: Named after the mathematician Gottfried Wilhelm Leibniz, this notation for partial derivatives employs differentials (dx and dy) to represent the variables and their infinitesimally small changes. In this notation, the partial derivative of z with respect to x can be written as ∂z/∂x or dz/dx.
Remember that the choice of notation varies, and different textbooks or disciplines may prefer one over the others. It's important to understand these notations to interpret and write mathematical expressions involving partial derivatives correctly.