By the chain rule:
( sin ( 2 x ) ) = cos ( 2 x ) · ( 2 x ) = 2 cos ( 2 x )
f ` ( sin 2 x ) = \(\lim_{h \to 0} \frac{sin(2x+2h)- sin(2x)}{h}= \\ \lim_{h \to 0} \frac{sin(2x)*sin(2h)+sin(2h)cos(2x)-sin(2x)}{h} = \\ \lim_{h \to 0} \frac{sin(2h)*cos(2x)}{h} = \\ \lim_{h \to 0} \frac{2sin(2h)*cos(2x)}{2h}=\)
= 2 cos ( 2 x )
because: \(\lim_{x \to 0} \frac{sinx}{x} =1\)
This is the reason why Δ x ( or h here ) is cancelled in the denominator
