Q87 / 180 · COMEDK 2014 · Mathematics
If $y=\tan x$, then $\frac{d^{2} y}{d x^{2}}=$
- A$1+y^{2}$
- Correct answerB$2 y\left(1+y^{2}\right)$
- C$y\left(1+y^{2}\right)$
- D$2 y\left(1-y^{2}\right)$
Solution
Given, $y=\tan x$
Differentiating w.r.t. $x$ both sides,
$$
\frac{d y}{d x}=\sec ^{2} x
$$
Taking again derivative w.r.t. $x$,
$$
\begin{aligned}
&\frac{d^{2} y}{d x^{2}}=2 \sec x \cdot \sec x \tan x \\
&=2 \sec ^{2} x \tan x=2 \tan x\left(1+\tan ^{2} x\right) \\
&=2 y\left(1+y^{2}\right)
\end{aligned}
$$
Hence, the correct answer is option: 'B'.