Question

If $x={\cos }^7\theta$ and $y=\sin \theta ,$ then $\frac{d^{3}x}{d{y}^3}=$

• A. $\frac{105}{4}\sin 4\theta$
• B. $\frac{105}{2}\sin 2\theta$
• C. $\frac{105}{4}\cos 4\theta$
• D. None of these

Answer 1

The correct option is A $\frac{105}{4}\sin 4\theta$

We have, $x={\cos }^7\theta$ and $y=\sin \theta$

Differentiating w.r.t $\theta$, we get

$\frac{dx}{d\theta }=-7{\cos }^6\theta \sin \theta$ and $\frac{dy}{d\theta }=\cos \theta$

Thus, we have $\frac{dx}{dy}=-7{\cos }^5\theta \sin \theta$

and $\frac{d^{2}x}{d{y}^2}=\frac{d}{dy}\left(\frac{dx}{dy}\right)=\frac{d}{d\theta }\left(\frac{dx}{dy}\right).\frac{d\theta }{dy}$

$=(35{\cos }^4\theta {\sin }^2\theta -7{\cos }^6\theta )\frac{1}{\cos \theta }=35{\cos }^3\theta {\sin }^2\theta -7{\cos }^5\theta$

$=35{\cos }^3\theta (1-{\cos }^2\theta )-7{\cos }^5\theta =35{\cos }^3\theta -42{\cos }^5\theta$

and, $\frac{d^{3}x}{d{y}^3}=\frac{d}{dy}\left(\frac{d^{2}x}{d{y}^2}\right)=\frac{d}{d\theta }\left(\frac{d^{2}x}{d{y}^2}\right).\frac{d\theta }{dy}$

$=(-105{\cos }^2\theta \sin \theta +210{\cos }^4\theta \sin \theta )\frac{1}{\cos \theta }$

$=105\sin \theta \cos \theta (2{\cos }^2\theta -1)=\frac{105}{2}\sin 2\theta \cos 2\theta =\frac{105}{4}\sin 4\theta$