If y=exp {sin 2x+sin 4x+sin 6x+……} then dydx=

y=esin2 x1-sin2 x =etan2 xDifferentiate both sidesdydx=etan2 x⋅2tan x⋅sec2 x =2⋅etan2 x⋅tan x⋅sec2 x

If y=exp {sin 2x+sin 4x+sin 6x+……} then dydx=

y=esin2 x1-sin2 x =etan2 xDifferentiate both sidesdydx=etan2 x⋅2tan x⋅sec2 x =2⋅etan2 x⋅tan x⋅sec2 x

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If $y = \exp {\sin 2 x + \sin 4 x + \sin 6 x + \dots \dots}$ then $\frac{d y}{d x} =$

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$e^{\sec 2 x}$

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

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

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

answer is C.

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Detailed Solution

$\begin{array}{rcl} y & = & e^{\frac{\sin^{2} x}{1 - \sin^{2} x}} \\ = & e^{\tan^{2} x} \end{array}$

Differentiate both sides

$\begin{array}{rcl} \frac{d y}{d x} & = & e^{\tan^{2} x} \cdot 2 \tan x \cdot \sec^{2} x \\ = & 2 \cdot e^{\tan^{2} x} \cdot \tan x \cdot \sec^{2} x \end{array}$

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If $y = \exp {\sin 2 x + \sin 4 x + \sin 6 x + \dots \dots}$ then $\frac{d y}{d x} =$