Answer:
D) [tex]4*ln(2)[/tex]
Step-by-step explanation:
Find the inverse function of f(x):
[tex]f(x)=log_2(2x)[/tex]
[tex]y=log_2(2x)[/tex]
[tex]x=log_2(2y)[/tex]
[tex]2^x=2y[/tex]
[tex]2^{x-1}=y[/tex]
[tex]f^{-1}(x)=2^{x-1}[/tex]
Take the derivative of the inverse function:
[tex]\frac{d}{dx}(f^{-1}(x))[/tex]
[tex]\frac{d}{dx}(2^{x-1})[/tex]
[tex]\frac{d}{dx}(e^{(x-1)(ln2)})[/tex] <-- Logarithmic Differentiation
[tex]ln(2)*e^{(x-1)(ln(2))[/tex]
[tex]ln(2)*2^{x-1}[/tex]
Substitute x=3:
[tex]ln(2)*2^{3-1}[/tex]
[tex]ln(2)*2^2[/tex]
[tex]4*ln(2)[/tex]
