The derivative of ln(2x) is 1/x. This is due to the rules of derived logarithmic expressions, which state that the derivative of ln(ax), where "a" is any real number, is equal to 1/x.
In order to prove the derivation of ln(ax), substitution and various derivatives need to be taken. For the sake of proof, let a = 1. Therefore, ln(ax) can be rewritten as ln(x). It is also necessary to use the function e^x, which is the inverse function of ln(x). According to the laws of inverse functions, when inverse functions are applied to one another, they cancel one another out. Therefore, e^ln(x) is equal to x. Since this function is somewhat long, it can be simplified by setting y equal to ln(x). This allows e^ln(x) to be rewritten as e^y. This function can now be differentiated. Taking the derivative of e^y is written as dy/dx e^y, which is equivalent to dy/dx x. By the Power Rule, the derivative of x is 1. Therefore dy/dx x = 1. Dividing by x on both sides yields dy/dx = 1/x. However, this can be further simplified, since y was previously set to equal ln(x). Therefore, d/dx ln(x) = 1/x.
This proof is semi-complicated since it uses multiple rules and Implicit differentiation. A simpler way to understand this may be to view a graph of ln(x) and 1/x. A graphical representation will show how 1/x is the derivative of ln(ax).