6. (2 pts) If \( k(x)=1.234 x^{4}+4.56 x^{2}+7.89 \) and \( f(k)=7-3 \ln k \), find \( \left.\frac{d f}{d x}\right|_{x=2} \) Use your calculator \( Y= \) list and \( n D e r i v \).

To find \( \left.\frac{d f}{d x}\right|_{x=2} \) using the chain rule, we first need to find the derivative of \( f(k) \) with respect to \( k \) and then multiply it by the derivative of \( k(x) \) with respect to \( x \). Given: \( k(x) = 1.234x^4 + 4.56x^2 + 7.89 \) \( f(k) = 7 - 3\ln k \) 1. Find the derivative of \( f(k) \) with respect to \( k \): \( \frac{df}{dk} = -\frac{3}{k} \) 2. Find the derivative of \( k(x) \) with respect to \( x \): \( \frac{dk}{dx} = 4.968x^3 + 9.12x \) 3. Apply the chain rule to find \( \frac{df}{dx} \): \( \frac{df}{dx} = \frac{df}{dk} \cdot \frac{dk}{dx} \) Substitute the values of \( \frac{df}{dk} \) and \( \frac{dk}{dx} \) into the chain rule formula: \( \frac{df}{dx} = -\frac{3}{k} \cdot (4.968x^3 + 9.12x) \) Now, substitute \( x = 2 \) into the expression to find \( \left.\frac{df}{dx}\right|_{x=2} \): \( \left.\frac{df}{dx}\right|_{x=2} = -\frac{3}{k(2)} \cdot (4.968(2)^3 + 9.12(2)) \) Calculate the value of \( \left.\frac{df}{dx}\right|_{x=2} \) using the given values. Calculate the value by following steps: - step0: Calculate: \(\frac{-3}{\left(1.234\times 2^{4}+4.56\times 2^{2}+7.89\right)}\times \left(4.968\times 2^{3}+9.12\times 2\right)\) - step1: Remove the parentheses: \(\frac{-3}{1.234\times 2^{4}+4.56\times 2^{2}+7.89}\times \left(4.968\times 2^{3}+9.12\times 2\right)\) - step2: Multiply the numbers: \(\frac{-3}{1.234\times 2^{4}+4.56\times 2^{2}+7.89}\times \left(39.744+9.12\times 2\right)\) - step3: Multiply the numbers: \(\frac{-3}{1.234\times 2^{4}+4.56\times 2^{2}+7.89}\times \left(39.744+18.24\right)\) - step4: Add the numbers: \(\frac{-3}{1.234\times 2^{4}+4.56\times 2^{2}+7.89}\times 57.984\) - step5: Multiply the numbers: \(\frac{-3}{19.744+4.56\times 2^{2}+7.89}\times 57.984\) - step6: Multiply the numbers: \(\frac{-3}{19.744+18.24+7.89}\times 57.984\) - step7: Add the numbers: \(\frac{-3}{45.874}\times 57.984\) - step8: Divide the numbers: \(-\frac{1500}{22937}\times 57.984\) - step9: Convert the expressions: \(-\frac{1500}{22937}\times \frac{7248}{125}\) - step10: Multiply the fractions: \(\frac{-1500\times 7248}{22937\times 125}\) - step11: Multiply the terms: \(\frac{-10872000}{22937\times 125}\) - step12: Multiply the terms: \(\frac{-10872000}{2867125}\) - step13: Multiply the numbers: \(-\frac{10872000}{2867125}\) - step14: Reduce the fraction: \(-\frac{86976}{22937}\) The derivative of \( f \) with respect to \( x \) at \( x = 2 \) is approximately -3.791952.