If y = e1/2 ˡᵒᵍ⁽¹⁺ᵗᵃⁿ²ˣ⁾, then dy/dx is equal to

The correct option is sec x tan x.

To find the derivative of the given function, let us first simplify the expression for $y$.
The given function is:
$$y=e^{\frac{1}{2}\log (1+{\tan }^{2}x)}$$

We know from trigonometric identities that:
$$1+{\tan }^{2}x={\sec }^{2}x$$
Substituting this identity into our equation for $y$, we get:
$$y=e^{\frac{1}{2}\log \left({\sec }^{2}x\right)}$$

Using the logarithmic property $a\log b=\log \left(b^a\right)$, we can rewrite the exponent:
$$\frac{1}{2}\log \left({\sec }^{2}x\right)=\log {\left({\sec }^{2}x\right)}^{1/2}=\log \left(\sec x\right)$$
Now, substitute this simplified exponent back into the expression for $y$:
$$y=e^{\log \left(\sec x\right)}$$

Since the exponential function and the natural logarithmic function are inverses of each other, $e^{\log u}=u$. Therefore, the function simplifies to:
$$y=\sec x$$

Now, we differentiate $y$ with respect to $x$:
$$\frac{dy}{dx}=\frac{d}{dx}\left(\sec x\right)$$
Using the standard derivative rule for the secant function, we get:
$$\frac{dy}{dx}=\sec x\tan x$$