Rules of total differentials
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:

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

When evaluating around $(X, Y, \dots) = (X_0, Y_0, \dots)$ (e.&thinsp;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:

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

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

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

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

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

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 $Y^v = (1 - t) Y$:

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

References:


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