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