#### Explain solution RD Sharma class 12 chapter 11 Higher Order Derivatives exercise very short answer type question 4 maths

\begin{aligned} &\text { The value of }\frac{d^{2} y}{d x^{2}} \text { at } x=\frac{1}{2} \end{aligned}

Hint:

First diff. $x$ &w.r.t $t$ .

\begin{aligned} \frac{d y}{d x}=& \frac{\frac{d y}{d t}}{\frac{d x}{d t}} \\ \end{aligned}

\begin{aligned} Di\! f\! f\; it\; w.r.t \; x\; to\; get \frac{d^{2} y}{d x^{2}} \end{aligned}

Given:

$x=2at \; and \; y=at^{2}$

Explanation:

It is given that

$x=2at \; and \; y=at^{2}$

Diff. $x$ w.r.t $t$

$\\\frac{d x}{d y}=2 a \\\\Di\! f\! f.\; w.r.t\; \; t \\\\\frac{d x}{d y}=2 a t \\\\Now,$

\begin{aligned} \frac{d y}{d x}=& \frac{\frac{d y}{d t}}{\frac{d x}{d t}} \\ =& \frac{2at}{2a} \\ =&t \end{aligned}

\begin{aligned} &\frac{d}{d x}\left(\frac{d y}{d x}\right)=\frac{d}{d x}(t) \\ &\frac{d^{2} y}{d x^{2}}=\frac{d t}{d x} \\ &\therefore \frac{d^{2} y}{d x^{2}}=\frac{1}{2 a} \\ &{\left[\because \frac{d x}{d t}=2 a\right]} \end{aligned}

So,

\begin{aligned} &\left(\frac{d^{2} y}{d x^{2}}\right)_{x=\frac{1}{2}}=\frac{1}{2 a} \\ &\text { Hence, } \frac{d^{2} y}{d x^{2}} \text { at } x=\frac{1}{2} \text { is } \frac{1}{2 a} . \end{aligned}