chain rule = logically inconsistent?

Question

I feel retarded asking this, since there must be something extremely obvious that I'm missing. But I can't think of anything wrong with my logic:

the derivative of sin(x) = cos(x)

therefore, the derivative of sin(5) = cos(5) [x = 5, in this case]
the derivative of sin(10) = cos(10) [x = 10, in this case]
the derivative of sin(20) = cos(20) [x = 20, in this case]
the derivative of sin(100) = cos(100) [x = 100, in this case]

according to the chain rule, the derivative of sin(2x) = 2cos(2x); we could then say:

the derivative of sin(5) = 2.5cos(5) [x = 2.5, in this case]
the derivative of sin(10) = 5cos(10) [x = 5, in this case]
the derivative of sin(20) = 10cos(20) [x = 10, in this case]
the derivative of sin(100) = 50cos(100) [x = 50, in this case]

This is inconsistent with the above. What gives?

Answer 1

You are making an implicit assumption in your reasoning that is unjustified. Specifically, you are assuming that if there exist a∈R and b∈R such that f(a) = g(b), then f'(a) = g'(b)

Of course, this is obviously false: consider f(x) = x and g(x) = 1. Then you have points a and b such that f(a) = g(b) (and indeed, a=b=1), but the derivatives of these two functions are obviously unequal (f'(1) = 1, g'(1) = 0)

The derivative is not a function from numbers to numbers, but a function from functions to functions. The derivative of sin x evaluated at 5 is not the same thing as the derivative of sin 5 (the former is cos 5, whereas the latter is 0, since sin 5 is a constant).

Now do you understand?

Answer 2

You're treating the cosine as if it were a linear function. It isn't.
If a function is linear, then f(cx) = cf(x)
Remember all the complicated rules for double angle and multple angle formulas in trig. You woudn't need these if the trig functions were linear.

Answer 3

I think you'll find the answer in that wonderful book by Dan Brown called "the Da Vinci Code"