It is given that $y = \tan[\ln(1+2x)]$. Show that \((1+2x)\frac{\mathrm{d}y}{\mathrm{d}x

Question

It is given that $y = \tan[\ln(1+2x)]$. Show that $(1+2x)\frac{\mathrm{d}y}{\mathrm{d}x} = 2(1+y^2)$.[2] By differentiating the result in part (a), find the Maclaurin expansion of $\tan[\ln(1+2x)]$ up to and including the term in $x^2$.[3] Hence, find the Maclaurin expansion of $\mathrm{e}^{\frac{1}{2}y}$, up to and including the term in $x^2$.[2] Using your answer in part (c), evaluate $\lim \limits_{x \to 0} \frac{\mathrm{e}^{\frac{1}{2}y}-1}{3x}$.[2]

Accepted Answer

(a) $(1+2x)\dfrac{\mathrm dy}{\mathrm dx}=2(1+y^2)$ (b) $y=2x-2x^2+O(x^3)$ (c) $\mathrm e^{y/2}=1+x-\dfrac12x^2+O(x^3)$ (d) $\dfrac13$

2024 EJC Promo Q8

2024 EJC Promo Q8

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