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Question Number 176809 by cortano1 last updated on 27/Sep/22

$$\{\begin{array}{l}\mathrm{x}=\cos {}^3\mathrm{\varnothing }\\ \mathrm{y}=\sin {}^3\mathrm{\varnothing }\end{array}\Rightarrow \frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}=?$$

Answered by mr W last updated on 27/Sep/22

$$\frac{dy}{d\varphi }=-3{\cos }^2\varphi \sin \varphi$$$$\frac{dx}{d\varphi }=3{\sin }^2\varphi \cos \varphi$$$$\Rightarrow \frac{dy}{dx}=\frac{\frac{dy}{d\varphi }}{\frac{dx}{d\varphi }}=\frac{-3{\cos }^2\varphi \sin \varphi }{3{\sin }^2\varphi \cos \varphi }=-\frac{1}{\tan \varphi }$$$$\frac{d^{2}y}{{dx}^2}=\frac{d}{dx}\left(\frac{dy}{dx}\right)=\frac{d}{d\varphi }\left(\frac{dy}{dx}\right)\times \frac{1}{\frac{dx}{d\varphi }}$$$$=\frac{1}{{\sin }^2\varphi }\times \frac{1}{3{\sin }^2\varphi \cos \varphi }$$$$=\frac{1}{3{\sin }^4\varphi \cos \varphi }$$

Commented by Tawa11 last updated on 28/Sep/22

$$\mathrm{Great}\mathrm{sirs}$$

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