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

$\begin{aligned} & y=e^{\sin ^{2} x+\sin ^{4} x+\sin ^{6} x+\cdots+\infty} \\ \Rightarrow \ln y & =\ln e^{\left(\sin ^{2} x+\sin ^{4} x+\sin ^{6} x+\cdots+\infty\right)} \\ \Rightarrow \ln y & =\sin ^{2} x+\sin ^{4} x+\sin ^{6} x+\cdots \infty \\ \Rightarrow & \ln y=\frac{\sin ^{2} x}{1-\sin ^{2} x} \Rightarrow \ln y=\frac{\sin ^{2} x}{\cos ^{2} x} \\ \Rightarrow & \ln y=\tan ^{2} x\end{aligned}$

Diff. w.r.t. $x$,

$\begin{array}{l} \frac{1}{y} \frac{d y}{d x}=2 \tan x\left(\sec ^{2} x\right) \\ \Rightarrow \frac{d y}{d x}=y\left(2 \tan x \cdot \sec ^{2} x\right) \\ \Rightarrow \frac{d y}{d x}=e^{\left(\sin ^{2} x+\sin ^{4} x+\sin ^{6} x+\cdots \infty\right)}\left(2 \tan x \cdot \sec ^{2} x\right) \\ \Rightarrow \frac{d y}{d x}=e^{\tan ^{2} x}\left(2 \tan x \sec ^{2} x\right) \\ \Rightarrow \frac{d y}{d x}=2 \tan x \sec ^{2} x \cdot e^{\tan ^{2} x} \\ \end{array}$