Question

What is the derivative of `2^x`?

Answer 1

`2^x*ln(2)`

Explanation:

Use the chain rule and the identity:

`d/(dt) e^t = e^t`

Start by using properties of exponents:

`2^x = (e^(ln 2))^x = e^(x ln 2)`

So if we put `t = x ln 2`, then:

`(dt)/(dx) = ln 2`

and:

`d/(dx) 2^x = d/(dx) e^(x ln 2) = (dt)/(dx) d/(dt) e^t = e^t * ln(2)`

`=e^(x ln 2)*ln(2)=2^x*ln(2)`