Derivative of y=e^lg2x

1. Let $u = \log{\left(2 x \right)}$.

2. The derivative of $e^{u}$ is itself.

3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(2 x \right)}$:

   1. Let $u = 2 x$.

   2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$.

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 2 x$:

      1. The derivative of a constant times a function is the constant times the derivative of the function.

         1. Apply the power rule: $x$ goes to $1$

         So, the result is: $2$

      The result of the chain rule is:

      $$\frac{1}{x}$$

   The result of the chain rule is:

   $$2$$

The answer is:

$$2$$