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Q.
If $x = a\, sin\theta$ and $y = b \,cos\theta$, then $\frac{d^{2}\,y}{dx^{2}}$ is equal to
Continuity and Differentiability
Solution:
Given that, $x = a\, sin\theta$ and $y = b \,cos\theta$
On differentiating $w.r.t$. $\theta$, we get
$\frac{dx}{d\theta}=a\,cos\,\theta$ and $\frac{dy}{d\theta}=-b\,sin\,\theta$
$\therefore \frac{dy}{dx}=\frac{dy / d\theta }{dx / d\theta}$
$=\frac{-b}{a}tan\,\theta$
Again differentiating $w.r.t. x$, we get
$\frac{d^{2}\,y}{dx^{2}}=\frac{-b}{a}sec^{2}\,\theta \frac{d\theta}{dx}$
$\Rightarrow \frac{d^{2}\,y}{dx^{2}}=-\frac{b}{a}sec^{2}\,\theta\left(\frac{1}{a\,cos\,\theta}\right)$
$=-\frac{b}{a^{2}}sec^{3}\,\theta$