CAIE A-Level Further Math A2 2.3.3 Maclaurin Series Questions

$\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{\frac{1}{2} \mathrm{e}^{x}}{1-\frac{1}{4} \mathrm{e}^{2 x}}$$\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}=\frac{\left(1-\frac{1}{4} \mathrm{e}^{2 x}\right)\left(\frac{1}{2} \mathrm{e}^{x}\right)-\frac{1}{2} \mathrm{e}^{x}\left(-\frac{1}{2} \mathrm{e}^{2 x}\right)}{\left(1-\frac{1}{4} \mathrm{e}^{2 x}\right)^{2}}$$f^{\prime}(0)=\frac{2}{3} \quad f^{\prime \prime}(0)=\frac{10}{9}$
Evaluates derivatives at x=0.

$f(0)=\tanh ^{-1}\left(\frac{1}{2}\right)=\frac{1}{2} \ln \left(\frac{3}{2} \times 2\right)$
Uses logarithmic form of $\tanh ^{-1}$.

$\frac{1}{2} \ln 3+\frac{2}{3} x+\frac{5}{9} x^{2}$
Applies $f(x)=f(0)+f^{\prime}(0) x+\frac{1}{2!} f^{\prime \prime}(0) x^{2}$

6