Total Derivative

Definition

When variables are related. For example:

(1) z = f(x, y)
Total derivative of z is:

Hint: to write z as a function f of variables x, y, and let the total derivative of z to be made of partial derivative of f and x, y, as well as dx, dy.

(2) y = f(x, w), x = g(w), then is the total derivative.

Calculation of total derivative

1. take total differential
2. Divide both sides by dw: (by Chain rule of derivative)

(3)
f(L, K) is the production function. For isoquant, we have f(L, K) = Q
take total derivative of f wrt L and K:

Along the isoquant, we have , so df = 0.

Note

Example

Here is calculated from implicit function.
Let