Rules of total differentials [closed]

Question

What are the rules of total differentials? I. e., given an arbitrary expression of which to take the total differential, what rules can be applied to arrive at the desired result, and how does one go about doing so?

Answer 1

Recall that the total differential $df$ of a function $f : \mathbb{R}^n \to \mathbb{R}$, $x = (x_1, \dots, x_n) \mapsto f(x)$, is defined as

$$df = \sum_{i = 1}^n \frac{ \partial f }{ \partial x_i } dx_i = \sum_{i = 1}^n f_{x_i} dx_i = \nabla f \cdot \left(dx_1, \dots, dx_n\right)^\top$$

Let $X$, $Y$, $Z$ be variables, and let $a$, $b$, $c$, $k$ and $n$ be constants. Then the following rules are valid:

$$\begin{alignat}{3} dk & = 0 && \qquad \text{(constant rule)} \\ d(cX^n) & = cnX^{n-1} dX && \qquad\text{(power rule)} \\ d(aX + bY) & = a\cdot dX + b \cdot dY && \qquad\text{(sum rule)} \\ d(XY) & = Y \cdot dX + X \cdot dY && \qquad\text{(product rule)} \\ d(XYZ) & = YZ \cdot dX + XZ \cdot dY + XY \cdot dZ && \qquad\text{(product rule for three factors)} \\ d\left( \frac{X}{Y} \right) & = \frac{ Y \cdot dX - X \cdot dY }{ Y^2 } && \qquad\text{(quotient rule)} \\ d(X^{-1}) & = -\frac{ dX }{ X^2 } && \qquad\text{(inverse rule)} \\ d(X \circ Y) & = X'(Y) \cdot dY && \qquad\text{(chain rule)} \end{alignat}$$

When evaluating around $(X, Y, \dots) = (X_0, Y_0, \dots)$ (e. g. around the steady state of a model), replace $X$ by $X_0$, $Y$ by $Y_0$ etc. in the above rules. For $X = X_0$ or $Y = Y_0$ respectively:

$$\begin{align} d(cX^n) \biggr\rvert_{X = X_0} & = cnX_0^{n-1} dX \\ d(X^{-1}) \biggr\rvert_{X = X_0} & = -\frac{ dX }{ X_0^2 } \\ d(X \circ Y) \biggr\rvert_{Y = Y_0} & = X'(Y_0) \cdot dY \end{align}$$

For $(X, Y) = (X_0, Y_0)$:

$$\begin{align} d(aX + bY) \biggr\rvert_{(X, Y) = (X_0, Y_0)} & = a\cdot dX + b \cdot dY \\ d(XY) \biggr\rvert_{(X, Y) = (X_0, Y_0)} & = Y_0 \cdot dX + X_0 \cdot dY \\ d\left( \frac{X}{Y} \right) \Biggr\rvert_{(X, Y) = (X_0, Y_0)} & = \frac{ Y_0 \cdot dX - X_0 \cdot dY }{ Y_0^2 } \end{align}$$

For ($X, Y, Z) = (X_0, Y_0, Z_0)$:

$$\begin{align} d(XYZ) \biggr\rvert_{(X, Y, Z) = (X_0, Y_0, Z_0)} & = Y_0Z_0 \cdot dX + X_0Z_0 \cdot dY + X_0Y_0 \cdot dZ \end{align}$$

As an example, consider the IS equation

$$Y = C((1 - t) \cdot Y) + I(i, E) + G$$

of an IS/LM model of a closed economy with a flat tax rate $t$. Take the total differential around the current state of the economy, using the rules above and setting $Y^v = (1 - t) Y$:

$$\begin{align} dY & = d\left[ C((1 - t) Y) + I(i, E) + G \right] \\ & = d\left[ C((1 - t) Y) \right] + d\left[ I(i, E) \right] + dG \\ & = C_{Y^v} d\left[ (1 - t) Y \right] + I_i \, di + I_E \, dE + dG \\ & = C_{Y^v} \left[ Y_0 d(1 - t) + (1 - t_0) \, dY \right] + I_i \, di + I_E \, dE + dG \\ & = C_{Y^v} \left[ Y_0 (-dt) + (1 - t_0) \, dY \right] + I_i \, di + I_E \, dE + dG \\ & = C_{Y^v} \left[ (1 - t_0) \, dY - Y_0 \, dt \right] + I_i \, di + I_E \, dE + dG \end{align}$$

References:

• Chiang, Alpha C. and Kevin Wainwright, Fundamental Methods of Mathematical Economics, 4th ed., McGraw-Hill 2005, chapter 8.