Questions: If f(2)=4 and f'(2)=3/4, use the formula (f^(-1))'(x)=1/f'(f^(-1)(x)) to find (f^(-1))'(4). (f^(-1))'(4)=

If f(2)=4 and f'(2)=3/4, use the formula (f^(-1))'(x)=1/f'(f^(-1)(x)) to find (f^(-1))'(4). (f^(-1))'(4)=
Transcript text: If $f(2)=4$ and $f^{\prime}(2)=\frac{3}{4}$, use the formula $\left(f^{-1}\right)^{\prime}(x)=\frac{1}{f^{\prime}\left(f^{-1}(x)\right)}$ to find $\left(f^{-1}\right)^{\prime}(4)$. \[ \left(f^{-1}\right)^{\prime}(4)= \]
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Solution

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Solution Steps

To find \(\left(f^{-1}\right)^{\prime}(4)\), we use the formula \(\left(f^{-1}\right)^{\prime}(x)=\frac{1}{f^{\prime}\left(f^{-1}(x)\right)}\). Since \(f(2) = 4\), it follows that \(f^{-1}(4) = 2\). We then substitute \(f^{-1}(4) = 2\) into the formula to get \(\left(f^{-1}\right)^{\prime}(4) = \frac{1}{f^{\prime}(2)}\). Given \(f^{\prime}(2) = \frac{3}{4}\), we can calculate \(\left(f^{-1}\right)^{\prime}(4)\).

Step 1: Identify the Inverse Function Derivative Formula

To find \(\left(f^{-1}\right)^{\prime}(4)\), we use the formula: \[ \left(f^{-1}\right)^{\prime}(x) = \frac{1}{f^{\prime}\left(f^{-1}(x)\right)} \]

Step 2: Determine \(f^{-1}(4)\)

Since \(f(2) = 4\), it follows that: \[ f^{-1}(4) = 2 \]

Step 3: Substitute into the Derivative Formula

Now, we substitute \(f^{-1}(4) = 2\) into the derivative formula: \[ \left(f^{-1}\right)^{\prime}(4) = \frac{1}{f^{\prime}(2)} \]

Step 4: Use the Given Derivative Value

We know that \(f^{\prime}(2) = \frac{3}{4}\). Therefore, we can calculate: \[ \left(f^{-1}\right)^{\prime}(4) = \frac{1}{\frac{3}{4}} = \frac{4}{3} \]

Final Answer

\[ \boxed{\left(f^{-1}\right)^{\prime}(4) = \frac{4}{3}} \]

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