What is difference between partial differentiation and total differentiation of the function of two or more variables with example?

Answer 1

total differentiation is closer to implicit differentiation although you are not solving for dy/dx. in other words:

the total derivative of f(x1,x2,...,xk) with respect to xn=

[df(x1,x2,...,xk)/dx1][dx1/dxn] + df(x1,x2,...,xk)/dx2[dx2/dxn]+...+df(x1,x2,...,xk)/dxn +[df(x1,x2,...,xk)/dxn+1][dxn+1/dxn]+...+[df(x1,x2,...,xk)/dxk][dxk/dxn]

however, the partial derivative is not this way.

the partial derivative of f(x1,x2,...,xk) with respect to xn is just that, can't be expanded.

The chain rule is not the same as total differentiation either. The chain rule is for partially differentiating f(x1,x2,...,xk) with respect to a variable not included in the explicit form. In other words, xn has to be considered a function of this variable for all integers n. so the total derivative is similar to the chain rule, but not the same.