Value after differentiating cos (sinx) is

The correct option is -sin (sinx).cosx.

To find the derivative of the function $y=\cos \left(\sin \right(x\left)\right)$ with respect to $x$, we need to apply the chain rule of differentiation.

The chain rule states that if we have a composite function $y=f\left(g\right(x\left)\right)$, then its derivative with respect to $x$ is given by:
$$\frac{dy}{dx}=f^{\text{'}}\left(g\right(x\left)\right)·g^{\text{'}}\left(x\right)$$

Here, our outer function is $f\left(u\right)=\cos \left(u\right)$, where $u=g\left(x\right)=\sin \left(x\right)$ is the inner function.

First, we differentiate the outer function with respect to its inner argument $u$:
$$\frac{d}{du}\left(\cos \right(u\left)\right)=-\sin \left(u\right)$$
Substituting $u=\sin \left(x\right)$ back, we get:
$$-\sin \left(\sin \right(x\left)\right)$$

Next, we differentiate the inner function $g\left(x\right)=\sin \left(x\right)$ with respect to $x$:
$$\frac{d}{dx}\left(\sin \right(x\left)\right)=\cos \left(x\right)$$

Finally, we multiply the derivative of the outer function by the derivative of the inner function to obtain the complete derivative:
$$\frac{dy}{dx}=-\sin \left(\sin \right(x\left)\right)·\cos \left(x\right)$$

This matches the correct option -sin (sinx).cosx.