Compute $\dfrac{d}{dw}[\sqrt{10w}]$.
Solution: Compute, where "C.R." denotes the chain rule
$$\begin{array}{ll}
\dfrac{d}{dw}[\sqrt{10w}] &= \dfrac{d}{dw} \left[(10w)^{\frac{1}{2}} \right] \\
&\stackrel{C.R.}{=} \dfrac{1}{2} (10w)^{-\frac{1}{2}} \dfrac{d}{dw} [10w] \\
&=5 (10w)^{-\frac{1}{2}} \\
&= \dfrac{5}{\sqrt{10w}}.
\end{array}$$
If that is hard to follow, it is because you don't yet comprehend the chain rule.
Let me rewrite it in terms of the chain rule theorem: here let $f(u)=\sqrt{u}$ so $f'(u)=\dfrac{1}{2\sqrt{u}}$ and $g(w)=10w$ so $g'(w)=10$. With this, $f(g(w))=\sqrt{10w}$. Thus by the chain rule formula,
$$\dfrac{d}{dx}[f(g(x))] = f'(g(x)) g'(x) = \dfrac{1}{2\sqrt{10w}} \cdot 10 =\dfrac{5}{\sqrt{10w}}.$$
