C1 2017 2 Edexcel

2

It helps to write:

\displaystyle y = x^{1/2} + 4x^{-1/2} + 4

Then:

\displaystyle \frac {\mathrm dy} {\mathrm dx} = \frac 1 2 x^{-1/2} - 2x^{-3/2} = \frac 1 {2 \sqrt x} - \frac 2 {x^{3/2}}

At x = 8, we have:

\begin{align*}\frac {\mathrm dy} {\mathrm dx} & = \frac 1 {2 \sqrt 8} - \frac 2 {8^{3/2}} \\&= \frac 1 {4 \sqrt 2} - \frac 2 {16 \sqrt 2} \\ & = \frac 1 {\sqrt 2} \left(\frac 1 4 - \frac 1 8\right) \\ & = \frac {\sqrt 2} 2 \left(\frac 1 8\right) \\ & = \frac {\sqrt 2} {16}\end{align*}

In particular a = \dfrac 1 {16}.

noting that 8^{3/2} = 8 \times \sqrt 8 = 8 \times (2 \sqrt 2) = 16 \sqrt 2.

1 Like

:sunglasses: Hero