How do I find the derivative of 2^x?

To find the derivative of any number to the power "x", such as 2^x, 5^x, or even 4.13^x, we first consider the general form a^x. We need to be a little creative here. We know that any variable y can be rewritten as e^lny.  If we then say that y = a^x then we can say that y= e^lnax. Note that this is because ln(a)^x=xlna. So that means y = a^x = e^{lna * x}. Now we want to find the derivative of a^x, or (e^{lna * x} )', which is lna e^x(lna). This is because lna is a definite number, and so we derivate this the same way we would e^3x (which would be 3e^3x). Now, if the derivative equals lna e^x(lna) we see that actually, e^x(lna) is equal to y, so we can rewrite this further as lnay. Since y = a^x we can simplify this finally to lnaa^x. That means that the derivative of a^x is a^xlna. This is the general form and should be remembered. So, (2^x)'= 2^xln2.