#### Please solve RD Sharma class 12 chapter Higher Order Derivatives exercise 11.1 question 45 maths textbook solution

$\frac{8\sqrt{2}}{\pi a}$

Hint:

You must know the derivative of cos and sin function

Given:

\begin{aligned} &x=a(\cos t+t \sin t) \\ &y=a(\sin t-t \cos t) \text { find } \frac{d^{2} y}{d x^{2}} \text { at } t=\frac{\pi}{4} \end{aligned}

Solution:

\begin{aligned} &x=a(\cos t+t \sin t) \\ &\frac{d x}{d t}=a(-\sin t+t \cos t+\sin t) \\ &=a t \cos t \\ &y=a(\sin t-t \cos t) \\ &\frac{d y}{d t}=a(\cos t+t \sin t-\cos t) \\ &=a t \sin t \end{aligned}

\begin{aligned} &\therefore \frac{d y}{d x}=\frac{d y}{d t} \times \frac{d t}{d x}=\frac{a t \sin t}{a t \cos t}=\tan t \\ &\text { Again } \frac{d^{2} y}{d x^{2}}=\sec ^{2} t \frac{d t}{d x}=\sec ^{2} t \times \frac{1}{a t \cos t} \end{aligned}

\begin{aligned} &=\frac{\sec ^{2} t}{a t} \\ &\left.\frac{d^{2} y}{d x^{2}}\right]_{t=\frac{\pi}{4}}=\frac{\sec ^{2} \frac{\pi}{4}}{a \frac{\pi}{4}}=\frac{2 \sqrt{2} \times 4}{a \pi} \\ &=\frac{8 \sqrt{2}}{\pi a} \end{aligned}