Use implicit differentiation to find the slope of the tangent line to the curve \( \frac{y}{x+8 y}=x^{9}+5 \) at the point \( \left(1, \frac{6}{-47}\right) \) slope \( = \)

Find the derivative with respect to \( x \) for \( \frac{y}{x+8y}=x^{9}+5 \). Find the first derivative by following steps: - step0: Find the derivative with respect to \(x\): \(\frac{y}{x+8y}=x^{9}+5\) - step1: Take the derivative: \(\frac{d}{dx}\left(\frac{y}{x+8y}\right)=\frac{d}{dx}\left(x^{9}+5\right)\) - step2: Calculate the derivative: \(\frac{x\frac{dy}{dx}-y}{\left(x+8y\right)^{2}}=\frac{d}{dx}\left(x^{9}+5\right)\) - step3: Calculate the derivative: \(\frac{x\frac{dy}{dx}-y}{\left(x+8y\right)^{2}}=9x^{8}\) - step4: Cross multiply: \(x\frac{dy}{dx}-y=\left(x+8y\right)^{2}\times 9x^{8}\) - step5: Simplify the equation: \(x\frac{dy}{dx}-y=9\left(x+8y\right)^{2}x^{8}\) - step6: Move the constant to the right side: \(x\frac{dy}{dx}=9\left(x+8y\right)^{2}x^{8}+y\) - step7: Divide both sides: \(\frac{x\frac{dy}{dx}}{x}=\frac{9\left(x+8y\right)^{2}x^{8}+y}{x}\) - step8: Divide the numbers: \(\frac{dy}{dx}=\frac{9\left(x+8y\right)^{2}x^{8}+y}{x}\) - step9: Expand the expression: \(\frac{dy}{dx}=\frac{9x^{10}+144x^{9}y+576y^{2}x^{8}+y}{x}\) Substitute \( x=1 \) and \( y=\frac{6}{-47} \) into the derivative \( \frac{dy}{dx}=\frac{9x^{10}+144x^{9}y+576y^{2}x^{8}+y}{x} \). Evaluate the expression by following steps: - step0: Evaluate: \(\frac{9x^{10}+144x^{9}y+576y^{2}x^{8}+y}{x}\) - step1: Substitute: \(\frac{9\times 1^{10}+144\times 1^{9}\times \frac{6}{-47}+576\left(\frac{6}{-47}\right)^{2}\times 1^{8}+\frac{6}{-47}}{1}\) - step2: Rewrite the fraction: \(\frac{9\times 1^{10}+144\times 1^{9}\times \frac{6}{-47}+576\left(-\frac{6}{47}\right)^{2}\times 1^{8}+\frac{6}{-47}}{1}\) - step3: Evaluate the power: \(\frac{9\times 1+144\times 1^{9}\times \frac{6}{-47}+576\left(-\frac{6}{47}\right)^{2}\times 1^{8}+\frac{6}{-47}}{1}\) - step4: Evaluate the power: \(\frac{9\times 1+144\times 1\times \frac{6}{-47}+576\left(-\frac{6}{47}\right)^{2}\times 1^{8}+\frac{6}{-47}}{1}\) - step5: Rewrite the fraction: \(\frac{9\times 1+144\times 1\times \left(-\frac{6}{47}\right)+576\left(-\frac{6}{47}\right)^{2}\times 1^{8}+\frac{6}{-47}}{1}\) - step6: Evaluate the power: \(\frac{9\times 1+144\times 1\times \left(-\frac{6}{47}\right)+576\left(-\frac{6}{47}\right)^{2}\times 1+\frac{6}{-47}}{1}\) - step7: Multiply: \(\frac{9+144\times 1\times \left(-\frac{6}{47}\right)+576\left(-\frac{6}{47}\right)^{2}\times 1+\frac{6}{-47}}{1}\) - step8: Multiply the terms: \(\frac{9-\frac{864}{47}+576\left(-\frac{6}{47}\right)^{2}\times 1+\frac{6}{-47}}{1}\) - step9: Multiply the terms: \(\frac{9-\frac{864}{47}+\frac{20736}{2209}+\frac{6}{-47}}{1}\) - step10: Rewrite the fraction: \(\frac{9-\frac{864}{47}+\frac{20736}{2209}-\frac{6}{47}}{1}\) - step11: Calculate: \(\frac{-\frac{273}{2209}}{1}\) - step12: Divide the terms: \(-\frac{273}{2209}\) The slope of the tangent line to the curve \( \frac{y}{x+8y}=x^{9}+5 \) at the point \( \left(1, \frac{6}{-47}\right) \) is \( -\frac{273}{2209} \).