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(7pts) Let
f(x)=2^(x)+x. Use the theorem on derivatives of inverses to find
(f^(-1))^(')(11).

( $7$ pts) Let $f(x)=2^{x}+x$ . Use the theorem on derivatives of inverses to find $\left(f^{-1}\right)^{\prime}(11)$ .

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Full solution

Q. (

7

pts) Let

f(x)=2^{x}+x

. Use the theorem on derivatives of inverses to find

\left(f^{-1}\right)^{\prime}(11)

.

Find Derivative of $f(x)$ : First, we need to find the derivative of $f(x)$ , which is $f'(x)$ . $\newline$ $f'(x) = \frac{d}{dx} [2^{x} + x]$
Calculate $f'(x)$ : Using the power rule and the fact that the derivative of $x$ is $1$ , we get: $\newline$ $f'(x) = 2^{x} \cdot \ln(2) + 1$
Apply Derivative of Inverses Theorem: Now, we use the theorem on derivatives of inverses which states that $(f^{-1})^\prime(y) = \frac{1}{f^\prime(f^{-1}(y))}$ . We need to find $(f^{-1})^\prime(11)$ , so we need to calculate $f^\prime(f^{-1}(11))$ .
Find $x$ for $f(x) = 11$ : We need to find the $x$ -value such that $f(x) = 11$ . This means solving $2^{x} + x = 11$ for $x$ .
Approximate $x$ Value: This equation isn't easy to solve algebraically, so we might need to use numerical methods or graphing to approximate the solution. $\newline$ Let's say we found that $x \approx 3$ (this is an approximation for the sake of the example).

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