Question

Based on the table of values for the
differentiable, invertible function f
and its derivative, evaluate
(f-1)'(2).
х
1
2
3
f(x)
5
3
2
f'(x)
-2
7
-4​

Answer 1

SOLUTION

TO DETERMINE

Based on the table of values for the differentiable, invertible function f and its derivative, evaluate

$\displaystyle\sf{( {f}^{ - 1} )'(2)}$

EVALUATION

We know that

$\displaystyle\sf{f(( {f}^{ - 1} )(x)) = x }$

Differentiating both sides with respect to x we get

$\displaystyle\sf{f'(( {f}^{ - 1} )(x)).(( {f}^{ - 1} )'(x)) = 1 }$

$\displaystyle\sf{ \implies \: ( {f}^{ - 1} )'(x) = \frac{1}{f'(( {f}^{ - 1} )(x))} }$

Putting x = 2 we get

$\displaystyle\sf{ \implies \: ( {f}^{ - 1} )'(2) = \frac{1}{f'(( {f}^{ - 1} )(2))} }$

$\displaystyle\sf{ \implies \: ( {f}^{ - 1} )'(2) = \frac{1}{f'(3)} } \: \: (\because \: f(3) = 2)$

$\displaystyle\sf{ \implies \: ( {f}^{ - 1} )'(2) = \frac{1}{ - 4} }$

$\displaystyle\sf{ \implies \: ( {f}^{ - 1} )'(2) = - \frac{1}{4} }$

FINAL ANSWER

$\boxed{ \: \: \displaystyle\sf{ \: ( {f}^{ - 1} )'(2) = - \frac{1}{4} } \: \: }$

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Answer 2

Answer:

: -1/4

Step-by-step explanation:

very Simple just concentrate ✌