Can some one explain to me where does the x come from in the Taylor series. In sin(x) = 1 + X/1! +...

Why is it x and if the function turned into sin (2x) does it affect that x?

If a function is infinitely differentiable, then that begins the Taylor series. Differentiable wrt to x.

Taylor series are an expansion and approximation to functions such as sin, e, etc.

f(x)=sin(x), \;\ f'(x)=cos(x), \;\ f''(x)=-sin(x), \;\ f'''(x)=-cos(x), \;\ f^{4}(x)=sin(x)

Then it starts over.

Yes, the 2x will make a difference:

sin(2x)=2x-\frac{4x^{3}}{3}+\frac{4x^{5}}{15}-\frac{8x^{7}}{315}+\frac{4x^{9}}{2835}-........................................

sin(2x)=\sum_{k=0}^{\infty}(-1)^{k}\frac{(2x)^{2k+1}}{(2k+1)!}

But does this method only work for certain functions (like trig and oiler's number) or can it work for let's say x^3??