If $x=at^2,\ y=2at$, then find $\dfrac{d^2y}{dx^2}$.

CBSE Class 12 · 2020 · DifferentiationShort Answer

x = a t^{2}, y = 2 a t

, then find

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

Answer: $- \frac{1}{2 a t ^{3}}$

Differentiate the parametric equations with respect to

t

\frac{d x}{d t} = 2 a t, \frac{d y}{d t} = 2 a

First derivative:

\frac{d y}{d x} = \frac{d y / d t}{d x / d t} = \frac{2 a}{2 a t} = \frac{1}{t}

For the second derivative, differentiate

\frac{d y}{d x}

w.r.t.

x

using the chain rule

\frac{d}{d x} = \frac{1}{d x / d t} \frac{d}{d t}

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

\frac{d ^{2} y}{d x ^{2}} = - \frac{1}{2 a t ^{3}}

CBSE Class 12 Board · 2020 · Subjective