Finding $d(sin(3x))/dx$ with the definition of derivative.

Small angle approximation, probably. $sin(x) approx x$ for $x$ small.

More rigorously, asymptotic equivalence. Note the well-known limit

$$lim_{x to 0} frac{sin(x)}{x} = 1$$

This gives the asymptotic equivalence $sin(x) sim x$ as $x to 0$. This means, within some circumstances, you can interchange $x$ and $sin(x)$ in limits where $x to 0$. (In this case, $x=3h/2$.)

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An alternative angle of approach as pointed out by player3236 in the comments is that they could have multiplied to get a cancellation as so:

$$require{cancel}frac{2cos (3x+frac32h) cancel{sin frac32h}}{h} cdot frac{frac 3 2 h}{cancel{sin frac 3 2 h}} = frac{2cos (3x+frac32h)frac32h}{h}$$