The derivative of 2e^x is 2e^x, with two being a constant. Any constant multiplied by a variable remains the same when taking a derivative. The derivative of e^x is e^x.
E^x is an exponential function. The base for this function is e, Euler's number. This is an irrational number and is approximately 2.71. The number "e" should be treated like any other numerical base such as two or three. If the exponential function has a numerical base "a", the function can be written y = a^x. The derivative of this function is dy/dx = (a^x)ln(a). For example, the derivative of y=2^x is dy/dx=(2^x)ln(2). Thus the derivative of e^x is (e^x)ln(e). The natural log of e, ln(e), is one. Thus the derivative simplifies to e^x.
If the function contains anything more complicated than an x in the exponent, it is necessary to use the chain rule. The derivative is found exactly the same as before, and then this derivative is multiplied by the derivative of the exponent. For example, if the exponent is 2x, the derivative of 2x is two. If the exponent is x^2, the derivative is 2x. For the function is y=2e^(2x), the derivative is dy/dx=(2e^2x)(2), which simplifies to dy/dx=4e^(2x).