Question

The Leibnitz notation for the chain rule is

Answer 1

For this derivative, you could set

$\boxed{u = x^2 - 4x}$

Then

$y = \sin(x^2-4x) = \sin(u)$

so that

$\boxed{\dfrac{\mathrm dy}{\mathrm du} = \cos(u)}$

and

$\boxed{\dfrac{\mathrm du}{\mathrm dx} = 2x - 4}$

Then the chain rule gives

$\dfrac{\mathrm dy}{\mathrm dx} = \dfrac{\mathrm dy}{\mathrm du}\dfrac{\mathrm du}{\mathrm dx} \\\\ \dfrac{\mathrm dy}{\mathrm dx} = \cos(u)(2x-4) \\\\ \boxed{\dfrac{\mathrm dy}{\mathrm dx} = (2x-4)\cos(x^2-4x)}$