Verify f has an inverse function. Then use the function f & the given real number a to find (f^-1)'(a). F(x)=sqrt(x-4); a =2

Question

dumbcow · Answer

to get inverse, flip the variables and solve for "y"
$$x = \sqrt{y-4}$$
$$x^2 = y-4$$
$$y = x^2 + 4$$

now for derivative of inverse
$$y' = 2x$$
plug in x=2
$$y'(2) = 4$$

anonymous · Answer

So from there would you go $$inverse f \prime (2)= \frac{ 1 }{ f \prime(4) }$$ or would 4 be the final answer?

dumbcow · Answer

no 4 would be the answer since they want the derivative of inverse and we already found the inverse :)

to do it the other way, you would just take derivative of f(x) and use relation above
$$(f^{-1})'(2) = \frac{1}{f'(8)}$$
2 = sqrt(x-4)    ----> x = 8

dumbcow · Answer

$$f'(x) = \frac{1}{2 \sqrt{x-4}}$$

anonymous · Answer

Okay. Thank you.