Question

How does a partial derivative differ from an ordinary derivative?

Answer 1

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.

Ex: consider `f=4x^2+3y+z`
partial derivative of f with respect to x is `f' = 8x`

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 `f'=8x+3(dy)/(dx)+(dz)/(dx)`