Question

What is the derivative of 2^x?

Answer 1

`2^xln2`.

Explanation:

we know `d/dx a^x=a^x lna`.so the ans.

Answer 2

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

Explanation:

By the definition of natural logarithm:

`e^(ln 2) = 2`

So:

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

Given that:

`d/(dx) e^x = e^x`

we can use the chain rule to deduce:

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

`color(white)(d/(dx) 2^x) = e^(x ln 2) * d/(dx) (x ln 2)`

`color(white)(d/(dx) 2^x) = e^(x ln 2) * ln 2`

`color(white)(d/(dx) 2^x) = 2^x * ln 2`