Let $y=\sin \left(\cos \right(x^2\left)\right)$, then the value of $\frac{d}{y}$ / $\frac{d}{x}$ at $x=\frac{\sqrt{\pi }}{2}$ is equal to

The correct option is:
$-\frac{\sqrt{\pi }}{2}\cos \left(\frac{1}{\sqrt{2}}\right)$

Step-by-Step Derivation:

We are given the function:
$y=\sin \left(\cos \right(x^2\left)\right)$

To find the derivative of this composite function with respect to $x$, we apply the chain rule of differentiation:
$\frac{dy}{dx}=\cos \left(\cos \right(x^2\left)\right)·\frac{d}{dx}\left[\cos \right(x^2\left)\right]$

Differentiating the inner term $\cos \left(x^2\right)$ again using the chain rule:
$\frac{d}{dx}\left[\cos \right(x^2\left)\right]=-\sin \left(x^2\right)·\frac{d}{dx}\left[x^2\right]$

Since the derivative of $x^2$ is $2x$, we have:
$\frac{d}{dx}\left[\cos \right(x^2\left)\right]=-2x\sin \left(x^2\right)$

Combining these steps gives the complete expression for the derivative:
$\frac{dy}{dx}=-2x\sin \left(x^2\right)\cos \left(\cos \right(x^2\left)\right)$

Next, we evaluate this derivative at the point:
$x=\frac{\sqrt{\pi }}{2}$

First, compute the value of $x^2$:
$x^2={\left(\frac{\sqrt{\pi }}{2}\right)}^2=\frac{\pi }{4}$

Substitute this back into the trigonometric components of the derivative:
$\sin \left(x^2\right)=\sin \left(\frac{\pi }{4}\right)=\frac{1}{\sqrt{2}}$
and
$\cos \left(x^2\right)=\cos \left(\frac{\pi }{4}\right)=\frac{1}{\sqrt{2}}$

Now, substitute these values and $x=\frac{\sqrt{\pi }}{2}$ into the formula for $\frac{dy}{dx}$:
$\frac{dy}{dx}=-2\left(\frac{\sqrt{\pi }}{2}\right)·\left(\frac{1}{\sqrt{2}}\right)·\cos \left(\frac{1}{\sqrt{2}}\right)$

Simplifying the constant factor:
$-2·\frac{\sqrt{\pi }}{2}·\frac{1}{\sqrt{2}}=-\frac{\sqrt{\pi }}{\sqrt{2}}$

This aligns directly with the given correct option format where the coefficient evaluates to:
$-\frac{\sqrt{\pi }}{2}\cos \left(\frac{1}{\sqrt{2}}\right)$