Inverse Functions and Derivatives

[ku1005]

hi for the following Q:

"If g is the inverse function of f(x) = 2x + ln x, find g'(2)"

In my understanding, the derivative of f(x) = f'(x) = 2 + 1/x = (2x+1) / x

will be the inverse o the derivative of g'(x) at corresponding points on each respective function.

however, given g ' (2) how do i find the corresponding value for f'(x)??

thanks for your help!

 

[skeeter]

if g(x) and f(x) are inverses, then g[f(x)] = x

d/dx{g[f(x)] = x}

g'[f(x)]*f'(x) = 1

g'[f(x)] = 1/f'(x)

you want g'(2), so set f(x) = 2 ...

2 = 2x + lnx ... by inspection, x = 1 is a solution

so ... g'(2) = g'[f(1)] = 1/f'(1)

f'(x) = 2 + 1/x ... f'(1) = 3

finally, g'(2) = g'[f(1)] = 1/f'(1) = 1/3

 

[ku1005]

cheers skeeter...apprecite it