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The function $f$ is defined by $f(x)=-2 x^{3}-x^{2}-4 x$ and $f(-1)=5$. If $g(x)=f^{-1}(x)$ , what is the value of $g^{\prime}(5)$ ?
Answer Attempt 1 out of 3
\[
g^{\prime}(5)=
\]
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Step 1 ：The question is asking for the derivative of the inverse function at a specific point. We know that the derivative of the inverse function at a point is the reciprocal of the derivative of the original function at the corresponding point.

Step 2 ：In this case, we need to find the derivative of the original function at the point that corresponds to \(g(5)\), which is \(-1\) since \(f(-1) = 5\).

Step 3 ：Let's calculate the derivative of the function \(f\) at \(x = -1\). The function \(f\) is given by \(f(x) = -2x^3 - x^2 - 4x\).

Step 4 ：The derivative of \(f\) is \(f'(x) = -6x^2 - 2x - 4\). Substituting \(x = -1\) into \(f'(x)\), we get \(f'(-1) = -8\).

Step 5 ：Finally, the derivative of the inverse function \(g\) at \(5\) is the reciprocal of \(f'(-1)\), which is \(-1/8\).

Step 6 ：So, the value of \(g'(5)\) is \(\boxed{-\frac{1}{8}}\).