Please Solve RD Sharma Class 12 Chapter 11 Higher Order Derivatives Exercise Multiple Choice Questions Maths Textbook Solution Question 14

(a)

Hint:

We must know about the derivative.

Given:

$x=f(t), y=g(t)$

Explanation:

$x=f(t), y=g(t)$

$\\ \frac{d x}{d t}=f^{\prime}(t), \frac{d y}{d t}=g^{\prime}(t) \$

$\ \frac{d y}{dt}\cdot{\frac{d t}{d x}}=\frac{g^{\prime}(t)}{f^{\prime}(t)} \\$

\begin{aligned} &\frac{d y}{d x}=\frac{g^{\prime}(t)}{f^{\prime}(t)} \end{aligned}

Differentiate on both sides,

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

$=\frac{d}{d t}\left(\frac{g^{\prime}(t)}{f^{\prime}(t)}\right) \times \frac{d t}{d x} \\$

$=\frac{f^{\prime}(t) g^{\prime \prime}(t)-g^{\prime}(t) f^{\prime \prime}(t)}{\left(f^{\prime}(t)\right)^{2}} \times \frac{1}{f^{\prime}(t)}$

$=\frac{f^{\prime}(t) g^{\prime \prime}(t)-g^{\prime}(t) f^{\prime \prime}(t)}{\left(f^{\prime}(t)\right)^{3}}$

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