How does a partial derivative differ from an ordinary derivative?
In partial differentiation, only the variable with respect to which the the function is being differentiated is considered as variable and other variables are considered as constants.
partial derivative of f with respect to x is
Here you only differentiate x and other variables, viz., y and z, are considered constants, so their derivatives are zero.
In ordinary differentiation, all the variables are differentiated with respect to the considered variable.
Ex: consider the above function
It's differentiation with respect to x is