How does a partial derivative differ from an ordinary derivative? When should we

Question

How does a partial derivative differ from an ordinary derivative? When should we use partial derivatives and when should we use ordinary derivatives?

Explanation / Answer

For a function of several variables, for example f(x,y,z), the partial derivatives are simply usual derivative in one of the variables when you keep the other variables fixed. For example, take f(x,y,z) = x^2 y^3 sin(y - z) Then for computing the partial derivative with respect to z, you fix x and y and you derivate the function of one variable: z --> x^2 y^3 sin(y - z) D_z f(x,y,z) = - x^2 y^3 cos(y - z) (just as if x and y were constants). For the other partial derivatives, you get: D_x f(x,y,z) = = 2 x y^3 sin(y - z) D_y f(x,y,z) = = 3 x^2 y^2 sin(y - z) + x^2 y^3 cos(y - z) The total derivative is when 2 of the variables are function of the third one. Suppose here the last two arguments of f(x,y,z) are functions of x: y(x) and z(x). Then you get a function of one variable: x --> F(x) = f(x,y(x),z(x)) and the total derivative is just the usual derivative (with respect to x) of the function F. It can be expressed in terms of the partial derivatives of f: dF/dx = (D_x f)(x,y(x),z(x)) + (D_y f)(x,y(x),z(x)) dy(x)/dx +(D_z f)(x,y(x),z(x)) dz(x)/dx

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