The derivative of 2sin(x) is 2cos(x). To compute this derivative, the derivatives of the trigonometric functions need to be memorized and the constant multiple rule applied.
The proofs for the trigonometric derivatives are fairly long, so it is easier to simply memorize the derivatives for the six basic trigonometric functions. The derivative of sin(x) is cos(x). The constant multiple rule states that the value of the derivative of a function multiplied by a constant is the derivative of the original function multiplied by the constant. To find the derivative of 2sin(x), the 2 is multiplied by the cos(x) to give 2cos(x).