Original question:
Posted by Lost One on Tuesday, March 5, 2013 at 12:04pm.
Let f(x) = (2/3)x^3 - 2x + 1 with a restricted domain of [1,infinity]. What is the value of (f-1)'(x) when x = 13?
•Calc - Steve, Tuesday, March 5, 2013 at 12:58pm
since x = f^-1(y)
and dx/dy = 1/y'
f' = 2x^2-2
(f^-1)'(13) = 1/f'(13) = 1/(2*169-2) = 1/336
Follow up question:
Okay, I see the logic of this now, but I have another problem. This is for a multichoice worksheet, and the four choices are 1/13, 13, 3, & 1/16. Which one is the correct choice?
To find the correct choice for (f-1)'(x) when x = 13, we need to substitute the value of x into the expression we found in the initial question.
The expression for f'(x) is 2x^2 - 2.
Substituting x = 13 into the expression, we get f'(13) = 2(13)^2 - 2 = 338 - 2 = 336.
Using the formula (f^-1)'(x) = 1/f'(x), we can plug in the value we found for f'(13) into the formula.
(f^-1)'(13) = 1/f'(13) = 1/336.
Therefore, the correct choice is 1/336.