inverse derivative shortcut and chain rule I know that when you have the equation y=ln(x), and you need to find the derivative, you can use the shortcut y'= 1/x.  My question is why, when using the shortcut, do you have to multiply by the derivative of x?  I'm aware it has something to do with the chain rule but I don't understand why.
Here's an example of my question:  If y=ln(5x), why doesn't y'=1/5x ?    

Because the rule is, for $u:\mathbb{R}\to]0,+\infty[$ a derivable function,
$$(\ln\circ u)'=\frac{u'}{u},$$ which is indeed an application of the chain rule
$$(v\circ u)'=u'\times v'\circ u$$ giving here
$$(\ln\circ u)'=u'\times\frac{1}{u}=\frac{u'}{u}.$$
In your example, $u$ is given by $u : x\mapsto5\times x$ and you derivative is
$$(ln(5x))'=\frac{5}{5\times x}=\frac{1}{x}$$ for $x\in]0,+\infty[.$
