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Find a formula for $f^{-1}(x)$ and $\left(f^{-1}\right)^{\prime}(x)$ if $f(x)=-\sqrt{6-x}$
\[
\begin{array}{l}
f^{-1}(x)= \\
\left(f^{-1}\right)^{\prime}(x)=
\end{array}
\]
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Answer
\(\boxed{(f^{-1})'(x) = 2x}\) is the derivative of the inverse function
Step 1 ：Replace the function notation with y: \(y = -\sqrt{6 - x}\)
Step 2 ：Swap x and y: \(x = -\sqrt{6 - y}\)
Step 3 ：Square both sides to get rid of the square root: \(x^2 = -(6 - y)\)
Step 4 ：Multiply both sides by -1 to get rid of the negative sign: \(-x^2 = 6 - y\)
Step 5 ：Solve for y to get the inverse function: \(y = 6 + x^2\)
Step 6 ：\(\boxed{f^{-1}(x) = 6 + x^2}\) is the inverse function
Step 7 ：Use the power rule for derivatives to find the derivative of the inverse function: \((f^{-1})'(x) = 2x\)
Step 8 ：\(\boxed{(f^{-1})'(x) = 2x}\) is the derivative of the inverse function
