Suppose f is a one-to-one function with f(2)=8 and f′(2)=4. What is the value of (f^−1)′(8)?

Verified step by step guidance

1. Step 1: Recall the formula for the derivative of the inverse function. If f is a one-to-one function and differentiable at a point, then the derivative of its inverse function at a point is given by: $(f^{-1}{)}^{\prime }(b)=\frac{1}{f^{\prime }(a)}$, where $f(a)=b$.
2. Step 2: Identify the given values in the problem. We know that $f(2)=8$ and $f^{\prime }(2)=4$.
3. Step 3: Match the given values to the formula. Here, $a=2$ and $b=8$, so we need to find $(f^{-1}{)}^{\prime }(8)$.
4. Step 4: Substitute the known values into the formula. Using $(f^{-1}{)}^{\prime }(b)=\frac{1}{f^{\prime }(a)}$, substitute $f^{\prime }(2)=4$ into the formula to find $(f^{-1}{)}^{\prime }(8)$.
5. Step 5: Calculate the expression $(f^{-1}{)}^{\prime }(8)=\frac{1}{4}$. This gives the value of the derivative of the inverse function at the point where $f(x)=8$.