Sea \( f(x)=\frac{3 x^{2}+1}{x+2} \) Encuentra \( f^{\prime}(-3) \)

Find the first order derivative with respect to \( x \) for \( \frac{3x^{2}+1}{x+2} \). Evaluate the derivative by following steps: - step0: Evaluate the derivative: \(\frac{d}{dx}\left(\frac{3x^{2}+1}{x+2}\right)\) - step1: Use differentiation rules: \(\frac{\frac{d}{dx}\left(3x^{2}+1\right)\times \left(x+2\right)-\left(3x^{2}+1\right)\times \frac{d}{dx}\left(x+2\right)}{\left(x+2\right)^{2}}\) - step2: Calculate: \(\frac{6x\left(x+2\right)-\left(3x^{2}+1\right)\times 1}{\left(x+2\right)^{2}}\) - step3: Calculate: \(\frac{6x\left(x+2\right)-\left(3x^{2}+1\right)}{\left(x+2\right)^{2}}\) - step4: Calculate: \(\frac{3x^{2}+12x-1}{\left(x+2\right)^{2}}\) - step5: Calculate: \(\frac{3x^{2}+12x-1}{x^{2}+4x+4}\) Substitute \( x=-3 \) into the expression \( \frac{3x^{2}+12x-1}{x^{2}+4x+4} \). Evaluate the expression by following steps: - step0: Evaluate: \(\frac{3x^{2}+12x-1}{x^{2}+4x+4}\) - step1: Substitute: \(\frac{3\left(-3\right)^{2}+12\left(-3\right)-1}{\left(-3\right)^{2}+4\left(-3\right)+4}\) - step2: Multiply the terms: \(\frac{3^{3}+12\left(-3\right)-1}{\left(-3\right)^{2}+4\left(-3\right)+4}\) - step3: Multiply the numbers: \(\frac{3^{3}-36-1}{\left(-3\right)^{2}+4\left(-3\right)+4}\) - step4: Multiply the numbers: \(\frac{3^{3}-36-1}{\left(-3\right)^{2}-12+4}\) - step5: Subtract the numbers: \(\frac{-10}{\left(-3\right)^{2}-12+4}\) - step6: Calculate: \(\frac{-10}{1}\) - step7: Divide the terms: \(-10\) La derivada de la función \( f(x)=\frac{3x^{2}+1}{x+2} \) es \( f'(x)=\frac{3x^{2}+12x-1}{x^{2}+4x+4} \). Al evaluar \( f'(-3) \), obtenemos que \( f'(-3)=-10 \).