Edexcel A-Level Mathematics A2 P4.5.1 Differentiation of Simple Functions Questions

$\frac{\mathrm{d} x}{\mathrm{~d} t}=\sec t \tan t$ and $\frac{\mathrm{d} y}{\mathrm{~d} t}=\sqrt{3} \sec ^{2}\left(t+\frac{\pi}{3}\right)$$\left(\frac{\mathrm{d} y}{\mathrm{~d} x}=\right) \frac{\mathrm{d} y}{\mathrm{~d} t} \div \frac{\mathrm{d} x}{\mathrm{~d} t}=\frac{\sqrt{3} \sec ^{2}\left(t+\frac{\pi}{3}\right)}{\sec t \tan t}$ oe
(3)

(a) Alt

$x=\sec t \quad y=\sqrt{3} \frac{\tan t+\sqrt{3}}{1-\sqrt{3} \tan t}$$\frac{\mathrm{d} x}{\mathrm{~d} t}=\sec t \tan t$ and $\frac{\mathrm{d} y}{\mathrm{~d} t}=\sqrt{3} \frac{(1-\sqrt{3} \tan t) \sec ^{2} t-(\tan t+\sqrt{3})\left(-\sqrt{3} \sec ^{2} t\right)}{(1-\sqrt{3} \tan t)^{2}}($ oe )
$\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{\mathrm{d} y}{\mathrm{~d} t} \div \frac{\mathrm{d} x}{\mathrm{~d} t}=\sqrt{3} \frac{(1-\sqrt{3} \tan t) \sec ^{2} t-(\tan t+\sqrt{3})\left(-\sqrt{3} \sec ^{2} t\right)}{\sec t \tan t(1-\sqrt{3} \tan t)^{2}}$ oe

$$t=\frac{\pi}{3}\Rightarrow x=2,\quad y=-3$$$$\left.\frac{dy}{dx}\right|_{t=\pi/3} =\frac{\sqrt3\sec^2(2\pi/3)}{\sec(\pi/3)\tan(\pi/3)} =2$$y+3=2(x-2)y=2x-7