Question

[20 PTS] Let f(x) = log2(2x). Which of the following is the value of d/dx(f^-1(3)) ?

Answer 1

D)
`4*ln(2)`

Explanation:

Find the inverse function of f(x):

`f(x)=log_2(2x)`

`y=log_2(2x)`

`x=log_2(2y)`

`2^x=2y`

`2^(x-1)=y`

`f^(-1)(x)=2^(x-1)`

Take the derivative of the inverse function:

`(d)/(dx)(f^(-1)(x))`

`(d)/(dx)(2^(x-1))`

`(d)/(dx)(e^((x-1)(ln2)))` <-- Logarithmic Differentiation

`ln(2)*e^{(x-1)(ln(2))`

`ln(2)*2^(x-1)`

Substitute x=3:

`ln(2)*2^(3-1)`

`ln(2)*2^2`

`4*ln(2)`