Question

If x=at², y=2at, then d²y/dx²=...........,Select Proper option from the given options.
(a) -1/t²
(b) 1/t²
(c) -1/2at³
(d) 1/2at³

Answer 1

Answer:

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Answer 2

$\bold{\frac{d^{2} y}{d x^{2}} = (c) -\frac{1}{2 a t^{3}}}$

Given:

$x=a t^{2}$

$y=2 a t$

To find:

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

Step-by-step explanation:

On differentiating ‘x’ wrt to ‘t’, we get,

$\frac{d x}{d t}=2 a t$

On differentiating ‘y’ wrt to ‘t’, we get,

$\frac{d y}{d t}=2 a$

Now, we need to apply the chain rule, which is:

$\Rightarrow \frac{d y}{d x}=\frac{d y / d t}{d x / d t}$

$\Rightarrow \frac{d y}{d x}=\frac{2 a}{2 a t}$

$\therefore \frac{d y}{d x}=\frac{1}{t}$

On differentiating above equation wrt to ‘x’, we get,

$\frac{d^{2} y}{d x^{2}}=\frac{d}{d x}\left(\frac{d y}{d x}\right)$

On applying chain rule, we get,

$\frac{d^{2} y}{d x^{2}}=\frac{d}{d t} \frac{d t}{d x} \left(\frac{d y}{d x}\right)$

On reciprocating the equation, we get,

$\frac{d^{2} y}{d x^{2}}=\frac{\frac{d}{d t}\left(\frac{1}{t}\right)}{\frac{d x}{d t}}$

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

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

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