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Let
f(x)=x^(3)+2x-1 and let
g be the inverse function of
f. Notice that
f(2)=11.

g^(')(11)=

Let $f(x)=x^{3}+2 x-1$ and let $g$ be the inverse function of $f$ . Notice that $f(2)=11$ . $\newline$ $g^{\prime}(11)=$

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Transformations of quadratic functions

Full solution

Q. Let

f(x)=x^{3}+2 x-1

and let

g

be the inverse function of

f

. Notice that

f(2)=11

.

\newline

g^{\prime}(11)=

Identify Inverse Function: Since $f(2)=11$ , we know that $g(11)=2$ because $g$ is the inverse of $f$ .
Calculate Derivative of Inverse: To find $g'(11)$ , we use the formula for the derivative of the inverse function: $g'(f(x)) = \frac{1}{f'(g(x))}.$
Find Derivative of Original Function: First, we need to find $f'(x)$ , which is the derivative of $f(x)=x^3+2x-1$ . $f'(x) = 3x^2 + 2$ .
Substitute into Derivative: Now we substitute $x$ with $g(11)$ , which is $2$ , into $f'(x)$ to get $f'(g(11))$ . $\newline$ $f'(g(11)) = f'(2) = 3(2)^2 + 2 = 12 + 2 = 14$ .
Calculate Final Derivative: Finally, we calculate $g'(11)$ using the formula $g'(f(x)) = \frac{1}{f'(g(x))}$ . $\newline$ $g'(11) = \frac{1}{f'(2)} = \frac{1}{14}.$

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