<p>It is given that&nbsp;\(y = \tan[\ln(1+2x)]\).</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Show that&nbsp;\((1+2x)\frac{\mathrm{d}y}{\mathrm{d}x} = 2(1+y^2)\).<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">By differentiating the result in part <strong>(a)</strong>, find the Maclaurin expansion of \(\tan[\ln(1+2x)]\) up to and including the term in \(x^2\).<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Hence, find the Maclaurin expansion of \(\mathrm{e}^{\frac{1}{2}y}\), up to and including the term in \(x^2\).<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Using your answer in part <strong>(c)</strong>, evaluate \(\lim \limits_{x \to 0} \frac{\mathrm{e}^{\frac{1}{2}y}-1}{3x}\).<span class="inline-marks">[2]</span></div></li></ol>

It is given that \(y = \tan[\ln(1+2x)]\).

Show that \((1+2x)\frac{\mathrm{d}y}{\mathrm{d}x} = 2(1+y^2)\).

By differentiating the result in part **(a)**, find the Maclaurin expansion of \(\tan[\ln(1+2x)]\) up to and including the term in \(x^2\).

Hence, find the Maclaurin expansion of \(\mathrm{e}^{\frac{1}{2}y}\), up to and including the term in \(x^2\).

Using your answer in part **(c)**, evaluate \(\lim \limits_{x \to 0} \frac{\mathrm{e}^{\frac{1}{2}y}-1}{3x}\).

2024 EJC Promo Q8

2024 EJC Promo Q8