Question

If $y=e^{{\sin }^{2}x+{\sin }^{4}x+{\sin }^{6}x+...\infty }$, then $\frac{dy}{dx}=$

• A. $e^{{\tan }^{2}x}$
• B. $e^{{\tan }^{2}x}{\sec }^{2}x$
• C. $2e^{{\tan }^{2}x}\tan x.{\sec }^{2}x$
• D. $1$

Answer 1

Answer: (C)

$2e^{{\tan }^{2}x}\tan x.{\sec }^{2}x$

If $y=e^z$

then $z={\sin }^{2}x+({\sin }^{2}x{)}^2+({\sin }^{2}x{)}^3+.......\infty$

Obviously ${\sin }^{2}x$ lies between $0$ and $1$.

$z$ is a geometric series with common ratio $r={\sin }^{2}x$

Hence $z=\frac{a}{1-r}$ (sum of infinite geometric series with $|r|<1$)

So $z=\frac{{\sin }^{2}x}{1-{\sin }^{2}x}={\tan }^{2}x$

So $y=e^{{\tan }^{2}x}$

Hence by chain rule

$\frac{dy}{dx}=\left(e^{{\tan }^{2}x}\right)\times \left(2\tan x\right)\times \left({\sec }^{2}x\right)$