If $x=a(\theta-\sin\theta),\ y=a(1+\cos\theta)$, find $\dfrac{d^{2}y}{dx^{2}}$.

$\dfrac{1}{4a}\csc^{4}\dfrac{\theta}{2}$

CBSE Class 12 · 2011 · DifferentiationLong Answer

x = a (θ - sin θ), y = a (1 + cos θ)

, find

\frac{d ^{2} y}{d x ^{2}}

Answer: $\frac{1}{4 a} csc^{4} \frac{θ}{2}$

Differentiate w.r.t.

θ

\frac{d x}{d θ} = a (1 - cos θ), \frac{d y}{d θ} = - a sin θ

First derivative:

\frac{d y}{d x} = \frac{- a sin θ}{a ( 1 - cos θ )} = \frac{- 2 sin \frac{θ}{2} cos \frac{θ}{2}}{2 sin ^{2} \frac{θ}{2}} = - cot \frac{θ}{2}

Second derivative:

\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d θ} (- cot \frac{θ}{2}) \cdot \frac{1}{d x / d θ} = \frac{1}{2} csc^{2} \frac{θ}{2} \cdot \frac{1}{a ( 1 - cos θ )}

Using

1 - cos θ = 2 sin^{2} \frac{θ}{2}

\frac{d ^{2} y}{d x ^{2}} = \frac{1}{4 a} csc^{4} \frac{θ}{2}

CBSE Class 12 Board · 2011 · Subjective