Let \( f(x)=4+\frac{6}{x}+\frac{4}{x^{2}} \) \( f^{\prime}(x)=\square \) \( f^{\prime}(3)=\square \) \( f^{\prime \prime}(x)=\square \) \( f^{\prime \prime}(3)=\square \)

Find the first order derivative with respect to \( x \) for \( 4+6/x+4/x^2 \). Evaluate the derivative by following steps: - step0: Evaluate the derivative: \(\frac{d}{dx}\left(4+\frac{6}{x}+\frac{4}{x^{2}}\right)\) - step1: Add the terms: \(\frac{d}{dx}\left(\frac{4x^{2}+6x+4}{x^{2}}\right)\) - step2: Use differentiation rules: \(\frac{\frac{d}{dx}\left(4x^{2}+6x+4\right)\times x^{2}-\left(4x^{2}+6x+4\right)\times \frac{d}{dx}\left(x^{2}\right)}{\left(x^{2}\right)^{2}}\) - step3: Calculate: \(\frac{\left(8x+6\right)x^{2}-\left(4x^{2}+6x+4\right)\times 2x}{\left(x^{2}\right)^{2}}\) - step4: Calculate: \(\frac{x^{2}\left(8x+6\right)-2x\left(4x^{2}+6x+4\right)}{x^{4}}\) - step5: Calculate: \(\frac{-6x^{2}-8x}{x^{4}}\) - step6: Rewrite the expression: \(-\frac{6x^{2}+8x}{x^{4}}\) - step7: Factor the expression: \(-\frac{x\left(6x+8\right)}{x^{4}}\) - step8: Reduce the fraction: \(-\frac{6x+8}{x^{3}}\) Substitute \( x=3 \) into the expression \( -\frac{6x+8}{x^{3}} \). Evaluate the expression by following steps: - step0: Evaluate: \(-\frac{6x+8}{x^{3}}\) - step1: Substitute: \(-\frac{6\times 3+8}{3^{3}}\) - step2: Multiply the numbers: \(-\frac{18+8}{3^{3}}\) - step3: Add the numbers: \(-\frac{26}{3^{3}}\) - step4: Simplify: \(-\frac{26}{27}\) Find the first order derivative with respect to \( x \) for \( -\frac{6x+8}{x^{3}} \). Evaluate the derivative by following steps: - step0: Evaluate the derivative: \(\frac{d}{dx}\left(-\frac{6x+8}{x^{3}}\right)\) - step1: Use differentiation rules: \(-\frac{\frac{d}{dx}\left(6x+8\right)\times x^{3}-\left(6x+8\right)\times \frac{d}{dx}\left(x^{3}\right)}{\left(x^{3}\right)^{2}}\) - step2: Calculate: \(-\frac{6x^{3}-\left(6x+8\right)\times 3x^{2}}{\left(x^{3}\right)^{2}}\) - step3: Calculate: \(-\frac{6x^{3}-3x^{2}\left(6x+8\right)}{x^{6}}\) - step4: Calculate: \(-\frac{-12x^{3}-24x^{2}}{x^{6}}\) - step5: Calculate: \(\frac{12x+24}{x^{4}}\) Substitute \( x=3 \) into the expression \( \frac{12x+24}{x^{4}} \). Evaluate the expression by following steps: - step0: Evaluate: \(\frac{12x+24}{x^{4}}\) - step1: Substitute: \(\frac{12\times 3+24}{3^{4}}\) - step2: Multiply the numbers: \(\frac{36+24}{3^{4}}\) - step3: Add the numbers: \(\frac{60}{3^{4}}\) - step4: Rewrite the expression: \(\frac{3\times 20}{3^{4}}\) - step5: Reduce the fraction: \(\frac{20}{3^{3}}\) - step6: Simplify: \(\frac{20}{27}\) The derivative of the function \( f(x)=4+\frac{6}{x}+\frac{4}{x^{2}} \) is \( f'(x)=-\frac{6x+8}{x^{3}} \). Substitute \( x=3 \) into \( f'(x) \), we get \( f'(3)=-\frac{26}{27} \) or approximately -0.962. The second derivative of the function is \( f''(x)=\frac{12x+24}{x^{4}} \). Substitute \( x=3 \) into \( f''(x) \), we get \( f''(3)=\frac{20}{27} \) or approximately 0.740.