CAIE A-Level Further Math A2 2.3.2 An Expression for X Y D Questions

$\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{\dot{y}}{\dot{x}}=\frac{2}{t^{\frac{1}{2}}-t^{-\frac{1}{2}}}$
Finds first derivative.

$\frac{\mathrm{d}}{\mathrm{d} t}\left(\frac{2}{t^{\frac{1}{2}}-t^{-\frac{1}{2}}}\right)=\frac{-2\left(\frac{1}{2} t^{-\frac{1}{2}}+\frac{1}{2} t^{-\frac{3}{2}}\right)}{\left(t^{\frac{1}{2}}-t^{-\frac{1}{2}}\right)^{2}}$
Differentiates $\frac{\mathrm{d} y}{\mathrm{~d} x}$ with respect to t.

$\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}=\frac{\mathrm{d}}{\mathrm{d} t}\left(\frac{2}{t^{\frac{1}{2}}-t^{-\frac{1}{2}}}\right) \times \frac{\mathrm{d} t}{\mathrm{~d} x}=\frac{-2\left(\frac{1}{2} t^{-\frac{1}{2}}+\frac{1}{2} t^{-\frac{3}{2}}\right)}{\left(t^{\frac{1}{2}}-t^{-\frac{1}{2}}\right)^{3}}=-\frac{t+1}{(t-1)^{3}}$
Applies chain rule. OE. Does not have to be simplified
for A1.

0<t<1
Accept -1<t<1. CWO.

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