Derivative with constant in the denominator I have the formular $\frac{d(y(t)*a)}{d(t*b)}$ with a and b being constants, which can be changed to $\frac{a}{b}\frac{d(y(t))}{d(t)}$. My question is why this is possible? For the numerator it's clear because when deriving a formular the constants don't change, but why can this rule also be applied for the denominator in a derivative?

Write $w=y(t)a$ and $z=tb$. Then $$\frac{dw}{dt}=a\frac{d(y(t))}{dt}.$$ Since $t=\frac{z}{b}$, we get $$\frac{dt}{dz}=\frac{1}{b}.$$ Hence, $$\frac{d(y(t)a)}{d(tb)}=\overbrace{\frac{dw}{dz}=\frac{dw}{dt}\cdot\frac{dt}{dz}}^{\text{Chain Rule}}=a \frac{d(y(t))}{dt}\cdot\frac{1}{b}=\frac{a}{b}\cdot\frac{d(y(t))}{dt}.$$