<p>Recall that the total differential <span class="math-container">$df$</span> of a function <span class="math-container">$f : \mathbb{R}^n \to \mathbb{R}$</span>, <span class="math-container">$x = (x_1, \dots, x_n) \mapsto f(x)$</span>, is defined as</p>

<p><span class="math-container">$$df = \sum_{i = 1}^n \frac{ \partial f }{ \partial x_i } dx_i = \sum_{i = 1}^n f_{x_i} dx_i = \nabla f \cdot \left(dx_1, \dots, dx_n\right)^\top$$</span></p>

<p>Let <span class="math-container">$X$</span>, <span class="math-container">$Y$</span>, <span class="math-container">$Z$</span> be variables, and let <span class="math-container">$a$</span>, <span class="math-container">$b$</span>, <span class="math-container">$c$</span>, <span class="math-container">$k$</span> and <span class="math-container">$n$</span> be constants. Then the following rules are valid:</p>

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

<p>When evaluating around <span class="math-container">$(X, Y, \dots) = (X_0, Y_0, \dots)$</span> (e.&thinsp;g. around the steady state of a model), replace <span class="math-container">$X$</span> by <span class="math-container">$X_0$</span>, <span class="math-container">$Y$</span> by <span class="math-container">$Y_0$</span> etc. in the above rules. For <span class="math-container">$X = X_0$</span> or <span class="math-container">$Y = Y_0$</span> respectively:</p>

<p><span class="math-container">$$\begin{align}
d(cX^n) \biggr\rvert_{X = X_0} &amp; = cnX_0^{n-1} dX \\
d(X^{-1}) \biggr\rvert_{X = X_0} &amp; = -\frac{ dX }{ X_0^2 } \\
d(X \circ Y) \biggr\rvert_{Y = Y_0} &amp; = X'(Y_0) \cdot dY
\end{align}$$</span></p>

<p>For <span class="math-container">$(X, Y) = (X_0, Y_0)$</span>:</p>

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

<p>For (<span class="math-container">$X, Y, Z) = (X_0, Y_0, Z_0)$</span>:</p>

<p><span class="math-container">$$\begin{align}
d(XYZ) \biggr\rvert_{(X, Y, Z) = (X_0, Y_0, Z_0)} &amp; = Y_0Z_0 \cdot dX + X_0Z_0 \cdot dY + X_0Y_0 \cdot dZ
\end{align}$$</span></p>

<p>As an example, consider the IS equation</p>

<p><span class="math-container">$$Y = C((1 - t) \cdot Y) + I(i, E) + G$$</span></p>

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

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

<p>References:</p>

<ul>
<li>Chiang, Alpha C. and Kevin Wainwright, <em>Fundamental Methods of Mathematical Economics</em>, 4th ed., McGraw-Hill 2005, chapter 8.</li>
</ul>

CC BY-SA 4.0
2019-08-24T10:08:26.920Z
2019-08-24T10:08:26.920Z
chsk
<p>What are the rules of total differentials? I.&thinsp;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?</p>

CC BY-SA 4.0
2019-08-25T02:45:12.770Z
This question should be in math.stackexchange, or it is likely already there. You could revise the question and make it more economics-specific such as “total differentials and their applications in economics”. The answer provided does indeed connect this but this question as it stands is about math
Brennan
CC BY-SA 4.0
2019-08-25T10:53:31.780Z
I'm voting to close this question as off-topic because it is about math, not economics.
Giskard
CC BY-SA 4.0
2019-08-26T14:07:23.770Z
Fair enough. I posted it here on the assumption that economists (students in particular, like me) would be more likely to search for / find this information here on this site. Feel free to edit, amend, close and / or move as you see fit.
chsk
<mathematical-economics>
Rules of total differentials
chsk
# Rules of total differentials

(Asked by: username_0 on 2019-08-24)

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?

**Tags:** mathematical-economics
> **Comments:**
> **username_1**: This question should be in math.stackexchange, or it is likely already there. You could revise the question and make it more economics-specific such as “total differentials and their applications in economics”. The answer provided does indeed connect this but this question as it stands is about math  
> **username_2**: I'm voting to close this question as off-topic because it is about math, not economics.  
> **username_0**: Fair enough. I posted it here on the assumption that economists (students in particular, like me) would be more likely to search for / find this information here on this site. Feel free to edit, amend, close and / or move as you see fit.  

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## 1 answers

### Answer by username_0 on 2019-08-24. Score: 3 | Accepted answer

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.


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