The Chain Rule may be stated informally as
$$\frac{\mathrm{d}z}{\mathrm{d}x}\overset{1}{=}\frac{\mathrm{d}z}{\mathrm{d}y}\cdot\frac{\mathrm{d}y}{\mathrm{d}x}.$$
Informal/intuitive interpretation:
$$\begin{array}{c} \text{The change in }z\text{ caused by}\\ \text{a small unit change in }x \end{array}\overset{2}{=}\begin{array}{c} \text{The change in }z\text{ caused by}\\ \text{a small unit change in }y \end{array}\times\begin{array}{c} \text{The change in }y\text{ caused by}\\ \text{a small unit change in }x. \end{array}$$
The Substitution Rule (integration) is the inverse/counterpart of the Chain Rule.
- How might we derive (even if only informally) the Substitution Rule from $\overset{1}{=}$?
- Is there an informal and intuitive interpretation of the Substitution Rule (similar to $\overset{2}{=}$ for the Chain Rule)?
Informally, you might think of the substitution rule this way:
\begin{align} &\begin{array}{c}\text{the accumulated}\\ \text{value due to} \\ \text{integration of}\end{array} \left( \begin{array}{c}\text{the rate of}\\ \text{accumulation per}\\ \text{unit change in $x$}\end{array} \times \begin{array}{c}\text{the small}\\ \text{change}\\ \text{in $x$}\end{array} \right)\\[2ex] &\qquad = \begin{array}{c}\text{the accumulated}\\ \text{value due to} \\ \text{integration of}\end{array}\left( \begin{array}{c}\text{the rate of}\\ \text{accumulation per}\\ \text{unit change in $u$}\end{array} \times \begin{array}{c}\text{the change in $u$}\\ \text{causedby a small} \\ \text{change in $x$}\end{array} \times \begin{array}{c}\text{the small}\\ \text{change}\\ \text{in $x$}\end{array} \right)\\[2ex] &\qquad = \begin{array}{c}\text{the accumulated}\\ \text{value due to} \\ \text{integration of}\end{array}\left( \begin{array}{c}\text{the rate of}\\ \text{accumulation per}\\ \text{unit change in $u$}\end{array} \times \begin{array}{c}\text{the small}\\ \text{change}\\ \text{in $u$}\end{array} \right)\\[2ex] \end{align}
When we write $\int f(x)\,dx = \int g(u)\, du$ the middle line is not explicitly included in the equation, but we invoke it implicitly when we write $f(x) = g(u) \frac{du}{dx}$ and $du = \frac{du}{dx} dx.$
Also implicitly assumed in all of this is that "the small change in $u$" is tied in a lockstep manner to "the small change in $x$" by defining $u$ as a function of $x$. (That is why I wrote "the small change" rather than "a small change". You cannot have just any small change in each place; each small change must be the change that corresponds to the small changes that occur the other parts of the equation.)
The relationship between the chain rule and the substitution rule is evident here.