This problem tests the skill of differentiating inverse functions. To find the derivative of the inverse function at a point b, use the formula $ (f^{-1})'(b) = \dfrac{1}{f'(a)} $, where a is the value such that $ f(a) = b $. In this case, $ f(2) = 32 $, so a = 2 and b = 32. Therefore, $ (f^{-1})'(32) = \dfrac{1}{f'(2)} $. A tempting distractor is choice B, $ \dfrac{1}{f'(32)} $, which incorrectly uses the derivative at b instead of at a. A transferable strategy for inverse derivatives is to identify the point a where $ f(a) = b $ and then compute the reciprocal of $ f'(a) $.

This problem tests the skill of differentiating inverse functions. To find the derivative of the inverse function at a point b, use the formula $(f^{-1})'(b) = 1 / f'(a)$, where a is the value such that $f(a) = b$. In this case, $f(0) = 1$, so a = 0 and b = 1. Therefore, $(f^{-1})'(1) = 1 / f'(0)$. A tempting distractor is choice A, $1/f'(1)$, which incorrectly uses the derivative at b instead of at a. A transferable strategy for inverse derivatives is to identify the point a where $f(a) = b$ and then compute the reciprocal of $f'(a)$.

To find the derivative of an inverse function, we apply $(f^{-1})'(b) = rac{1}{f'(a)}$ where $f(a) = b$. Given $f(3) = 2$, we need $(f^{-1})'(2) = rac{1}{f'(3)}$. For $f(x) = sqrt{x+1}$, we have $f'(x) = rac{1}{2sqrt{x+1}}$. Evaluating at $x = 3$: $f'(3) = rac{1}{2sqrt{4}} = rac{1}{4}$. Therefore, $(f^{-1})'(2) = 4$. Students might forget the chain rule when differentiating $sqrt{x+1}$ or evaluate at the wrong point. The key is recognizing that inverse derivatives reciprocate the original function's rate of change.

This problem requires finding the derivative of an inverse function at a point. Using $(f^{-1})'(b) = rac{1}{f'(a)}$ where $f(a) = b$, and knowing $f(0) = 0$, we need $(f^{-1})'(0) = rac{1}{f'(0)}$. Taking the derivative: $f'(x) = cos x + 1$, we get $f'(0) = cos(0) + 1 = 1 + 1 = 2$. Thus $(f^{-1})'(0) = rac{1}{2}$. A common mistake is computing $f'$ at the wrong point or forgetting that $cos(0) = 1$, not 0. Remember that the inverse function derivative formula creates a reciprocal relationship between the rates of change.

This problem tests the skill of differentiating inverse functions. To find the derivative of the inverse at a point, use the formula $(f^{-1}$)'(b) = 1 / f'(a), where f(a) = b. Here, a = 2 and b = 32, so compute f'(x) = $5x^4$, and f'(2) = 80. Thus, $(f^{-1}$)'(32) = 1/80. A tempting distractor is 80, which is f'(2) instead of its reciprocal. Always remember to take the reciprocal of the original function's derivative at the input point to find the inverse's derivative at the output point.

This problem tests the skill of differentiating inverse functions. To find the derivative of the inverse at a point, use the formula $(f^{-1}$)'(b) = 1 / f'(a), where f(a) = b. Here, a = 2 and b = 6, so compute f'(x) = $3x^2$ - 1, and f'(2) = 11. Thus, $(f^{-1}$)'(6) = 1/11. A tempting distractor is 11, which is f'(2) instead of its reciprocal. Always remember to take the reciprocal of the original function's derivative at the input point to find the inverse's derivative at the output point.

This problem tests the skill of differentiating inverse functions. To find the derivative of the inverse at a point, use the formula $(f^{-1}$)'(b) = 1 / f'(a), where f(a) = b. Here, a = 5 and b = 3, so compute f'(x) = 1/(2√(x+4)), and f'(5) = 1/6. Thus, $(f^{-1}$)'(3) = 6. A tempting distractor is 1/6, which is f'(5) instead of its reciprocal. Always remember to take the reciprocal of the original function's derivative at the input point to find the inverse's derivative at the output point.

This problem requires finding the derivative of an inverse function at a specific point. Using $(f^{-1})'(b) = rac{1}{f'(a)}$ where $f(a) = b$, and given $f(0) = 0$, we need $(f^{-1})'(0) = rac{1}{f'(0)}$. Computing $f'(x) = 5x^4 + 1$, we get $f'(0) = 5(0)^4 + 1 = 1$. Thus $(f^{-1})'(0) = rac{1}{1} = 1$. A tempting error is to think that since $f(0) = 0$, the derivative must also be 0, but this confuses function values with derivative values. Remember that the inverse derivative formula always involves taking a reciprocal of the original function's derivative.

This problem tests the skill of differentiating inverse functions. To find the derivative of the inverse function at a point b, use the formula $(f^{-1}$)'(b) = 1 / f'(a), where a is the value such that f(a) = b. In this case, f(1) = 4, so a = 1 and b = 4. Therefore, $(f^{-1}$)'(4) = 1 / f'(1). A tempting distractor is choice B, 1/f'(4), which incorrectly uses the derivative at b instead of at a. A transferable strategy for inverse derivatives is to identify the point a where f(a) = b and then compute the reciprocal of f'(a).

This problem tests the skill of differentiating inverse functions. To find the derivative of the inverse function at a point $b$, use the formula $(f^{-1})'(b) = \frac{1}{f'(a)}$, where $a$ is the value such that $f(a) = b$. In this case, $f(0) = 3$, so $a = 0$ and $b = 3$. Therefore, $(f^{-1})'(3) = \frac{1}{f'(0)}$. A tempting distractor is choice E, $\frac{1}{f'(3)}$, which incorrectly uses the derivative at $b$ instead of at $a$. A transferable strategy for inverse derivatives is to identify the point $a$ where $f(a) = b$ and then compute the reciprocal of $f'(a)$.