Question

If $x=a{t}^2,y=2at$, then $\frac{d^{2}y}{d{x}^2}=$

• A. $\frac{-1}{t^2}$
• B. $\frac{-1}{t^3}$
• C. $\frac{-1}{2a{t}^3}$
• D. $\frac{1}{2a{t}^3}$

Answer 1

The correct option is C

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

Explanation for the correct answer:

To find the value of $\frac{d^{2}y}{d{x}^2}$ :

Given,

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

Then, differentiate $x$ and $y$ with respect to $x$.

$\begin{array}{rcl}\frac{dx}{dt}& =& 2at\\ \frac{dy}{dt}& =& 2a\\ \frac{dy}{dx}& =& \frac{\frac{dy}{dt}}{\frac{dx}{dt}}\\ & =& \frac{2a}{2at}\\ & =& \frac{1}{t}\end{array}$

Now, differentiate the obtained derivative with respect to $x$

$\begin{array}{rcl}\frac{d^{2}y}{d{x}^2}& =& -\frac{1}{t^2}.\frac{dt}{dx}\\ & =& \left(-\frac{1}{t^2}\right)\times \left(\frac{1}{2at}\right)\\ & =& -\frac{1}{2a{t}^3}\end{array}$

Hence, the correct option is C.