AP Calculus AB / Differentiation Composite, Implicit, and Inverse Functions / Question No. 24

24. If $f (x) = 3 x^{3} + 1$ and $g$ is the inverse function of $f$ , what is the value of $g^{'} (25)$ ?

\frac{1}{9}

\frac{1}{81}

\frac{1}{36}

\frac{1}{54}

Answer is: option3

\frac{1}{36}

Solution:

To solve this, we use the formula for the derivative of the inverse function: $g^{'} (x) = \frac{1}{f^{'} (g (x))}$ We are asked to find $g^{'} (25)$ . First, we need to find $g (25)$ . Since $g$ is the inverse of $f$ , we know that $f (g (25)) = 25$ . Start by solving for $x$ such that: $f (x) = 25$ Given that $f (x) = 3 x^{3} + 1$ , we solve: $3 x^{3} + 1 = 25 ⟹ 3 x^{3} = 24 ⟹ x^{3} = 8 ⟹ x = 2$ Thus, $g (25) = 2$ . Now, we need to compute $f^{'} (x)$ , the derivative of $f (x)$ . Since $f (x) = 3 x^{3} + 1$ , we have: $f^{'} (x) = 9 x^{2}$ So, $f^{'} (2) = 9 \times (2)^{2} = 9 \times 4 = 36$ Finally, applying the formula for the inverse function's derivative: $g^{'} (25) = \frac{1}{f^{'} (g (25))} = \frac{1}{f^{'} (2)} = \frac{1}{36}$

The correct answer is C) $\frac{1}{36}$ .

Previous Next

AP Calculus AB Tags

AP Calculus BC Tags