compute \\( \\frac { d y } { d x } \\) using the chain rule. state your answer in terms of \\( x \\) only…

Explanation:

Step1: Find (\frac{dy}{du})

Differentiate (y = \frac{u}{4}+\frac{4}{u}) with respect to (u). Using the power rule (\frac{d}{du}(au^{n})=nau^{n - 1}), where (a=\frac{1}{4},n = 1) for the first - term and (a = 4,n=-1) for the second - term. (\frac{dy}{du}=\frac{1}{4}-\frac{4}{u^{2}})

Step2: Find (\frac{du}{dx})

Differentiate (u=x - x^{9}) with respect to (x). Using the power rule (\frac{d}{dx}(ax^{n})=nax^{n - 1}), where (a = 1,n = 1) for the first - term and (a=1,n = 9) for the second - term. (\frac{du}{dx}=1-9x^{8})

Step3: Apply the chain rule (\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx})

Substitute (\frac{dy}{du}=\frac{1}{4}-\frac{4}{u^{2}}) and (\frac{du}{dx}=1 - 9x^{8}) into the chain - rule formula. (\frac{dy}{dx}=\left(\frac{1}{4}-\frac{4}{u^{2}}\right)(1 - 9x^{8})) Since (u=x - x^{9}), substitute (u) into the expression: [ \begin{align*} \frac{dy}{dx}&=\left(\frac{1}{4}-\frac{4}{(x - x^{9})^{2}}\right)(1 - 9x^{8})\ &=\frac{1-9x^{8}}{4}-\frac{4(1 - 9x^{8})}{(x - x^{9})^{2}} \end{align*} ]

Answer:

(\frac{1-9x^{8}}{4}-\frac{4(1 - 9x^{8})}{(x - x^{9})^{2}})