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Differentiation

### Proving the Chain Rule

In this post, the chain rule is proved. This rule frequently appears in Calculus. Recall from this post that: $dx|_{a^+} \equiv \lim_{\Delta x \rightarrow 0^+} \Delta x$ and $df(x)|_{a^+} = \lim_{\Delta x \rightarrow 0^+} \Delta f (\Delta x)$. Suppose a variable $y$ can be written as a function of another variable $u$, and that $u$ can be written as a function of another variable $x$. Then $y$ can be written as a function of $x$: $y(x)$. What is $\frac{dy(x)}{dx}\bigg|_{a^+}$ if only $y(u)$ and $u(x)$ are known? Solution: Start by writing $\frac{dy(x)}{dx}\bigg|_{a^+} = \frac{dy(x)}{dx}\bigg|_{a^+}$. According to this post, the $|_{a^+}$ symbols can be moved as …