Question

How do you differentiate `f(x)= sqrt (ln2^x)`?

Answer 1

Use properties of `ln` and power rule to find:

`d/(dx) sqrt(ln 2^x) = 1/2 sqrt(ln 2^(1/x))`

Explanation:

Use properties of `ln` to note that `ln(2^x) = x ln(2)`.

Use power rule to differentiate `x^(1/2)`

Reformulate in terms of `2^(1/x)`

So:

`d/(dx) f(x) = d/(dx) sqrt(ln 2^x)`

`= d/(dx) sqrt(x ln(2))`

`=d/(dx) (x ln(2))^(1/2)`

`= sqrt(ln(2)) d/(dx) x^(1/2)`

`= sqrt(ln(2)) * 1/2 x ^(-1/2)`

`= 1/2sqrt(ln(2)/x)`

`= 1/2 sqrt(ln 2^(1/x))`