If $x=a\sin t$ and $y=a\!\left(\cos t+\log\tan\dfrac{t}{2}\right)$, find $\dfrac{d^{2}y}{dx^{2}}$.

CBSE Class 12 · 2013 · DifferentiationLong Answer

x = a sin t

and

y = a (cos t + lo g tan \frac{t}{2})

, find

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

Answer: $- \frac{csc ^{2} t}{a cos t}$

Differentiate w.r.t.

t

. For

y

, note

\frac{d}{d t} lo g tan \frac{t}{2} = \frac{1}{sin t}

\frac{d y}{d t} = a (- sin t + \frac{1}{sin t}) = a \frac{cos ^{2} t}{sin t}, \frac{d x}{d t} = a cos t

First derivative:

\frac{d y}{d x} = \frac{a cos ^{2} t / sin t}{a cos t} = \frac{cos t}{sin t} = cot t

Second derivative:

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

\frac{d ^{2} y}{d x ^{2}} = - \frac{csc ^{2} t}{a cos t}

CBSE Class 12 Board · 2013 · Subjective