Let \( f(x)=3+\frac{5}{x}+\frac{3}{x^{2}} \) \( f^{\prime}(x)=-\left((5 x+6) / x^{\wedge} 3\right) \) \( f^{\prime}(3)=5 \) \( f^{\prime \prime}(x)=\square \) \( f^{\prime \prime}(3)=\square \)

Given: \( f(x)=3+\frac{5}{x}+\frac{3}{x^{2}} \) \( f^{\prime}(x)=-\left(\frac{5x+6}{x^{3}}\right) \) \( f^{\prime}(3)=5 \) To find: \( f^{\prime \prime}(x) \) \( f^{\prime \prime}(3) \) First, let's find the second derivative of the function \( f(x) \) using the given first derivative \( f^{\prime}(x) \). Given: \( f^{\prime}(x)=-\left(\frac{5x+6}{x^{3}}\right) \) To find the second derivative \( f^{\prime \prime}(x) \), we need to differentiate \( f^{\prime}(x) \) with respect to \( x \). Let's calculate the second derivative \( f^{\prime \prime}(x) \) using the given first derivative \( f^{\prime}(x) \). Evaluate the derivative by following steps: - step0: Evaluate the derivative: \(\frac{d}{dx}\left(-\left(\frac{5x+6}{x^{3}}\right)\right)\) - step1: Remove the parentheses: \(\frac{d}{dx}\left(-\frac{5x+6}{x^{3}}\right)\) - step2: Use differentiation rules: \(-\frac{\frac{d}{dx}\left(5x+6\right)\times x^{3}-\left(5x+6\right)\times \frac{d}{dx}\left(x^{3}\right)}{\left(x^{3}\right)^{2}}\) - step3: Calculate: \(-\frac{5x^{3}-\left(5x+6\right)\times 3x^{2}}{\left(x^{3}\right)^{2}}\) - step4: Calculate: \(-\frac{5x^{3}-3x^{2}\left(5x+6\right)}{x^{6}}\) - step5: Calculate: \(-\frac{-10x^{3}-18x^{2}}{x^{6}}\) - step6: Calculate: \(\frac{10x+18}{x^{4}}\) The second derivative of the function \( f(x) \) is: \( f^{\prime \prime}(x) = \frac{10x+18}{x^{4}} \) Now, we are given that \( f^{\prime}(3) = 5 \). We can use this information to find \( f^{\prime \prime}(3) \). Substitute \( x = 3 \) into the second derivative expression \( f^{\prime \prime}(x) \) to find \( f^{\prime \prime}(3) \). Evaluate the expression by following steps: - step0: Evaluate: \(\frac{10x+18}{x^{4}}\) - step1: Substitute: \(\frac{10\times 3+18}{3^{4}}\) - step2: Multiply the numbers: \(\frac{30+18}{3^{4}}\) - step3: Add the numbers: \(\frac{48}{3^{4}}\) - step4: Rewrite the expression: \(\frac{3\times 16}{3^{4}}\) - step5: Reduce the fraction: \(\frac{16}{3^{3}}\) - step6: Simplify: \(\frac{16}{27}\) The second derivative of the function \( f(x) \) evaluated at \( x = 3 \) is: \( f^{\prime \prime}(3) = \frac{16}{27} \) or approximately \( 0.\dot{5}9\dot{2} \)