Question

If $x=a(\cos \theta +\sin \theta )$ and $y=a(\sin \theta -\cos \theta )$, then find $\frac{d^{2}y}{d{x}^2}$.

Answer 1

$x=acos\theta +asin\theta$

$\frac{dx}{d\theta }=-asin\theta +acos\theta$

$\frac{d^{2}x}{d{\theta }^2}-acos\theta -asin\theta$

$y=asin\theta -a\theta cos\theta$

$\frac{dy}{d\theta }=acos\theta -acos\theta +a\theta sin\theta$

$\frac{d^{2}y}{d\theta 2}=-asin\theta +asin\theta +asin\theta +a\theta cos\theta$

$\frac{d^{2}y}{d{x}^2}=\frac{d^{2}y/d{\theta }^2}{d{x}^2/d{\theta }^2}=\frac{-a(cos\theta +sin\theta )}{a(sin\theta +a\theta cos\theta )}$

$\frac{d^{2}y}{d{x}^2}=-\frac{(cos\theta +sin\theta )}{(sin\theta +a\theta cos\theta )}$