2021 SAJC Promo Q5
(a)
(i) Show that $\frac{\text{d}}{\text{d}x}\left( {{3}^{2x}} \right)={{9}^{x}}\ln 9$.
(ii) Hence, differentiate ${{\tan }^{-1}}\left( {{3}^{2x}} \right)$, $x>0$ with respect to $x$.
(b)
Given that ${{y}^{2y}}={{e}^{3x+2e}}$, where $y>{{e}^{-1}}$, show that $y\frac{{{\text{d}}^{2}}y}{\text{d}{{x}^{2}}}=-\frac{2}{3}{{\left( \frac{\text{d}y}{\text{d}x} \right)}^{3}}$.
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