Value after differentiating cos (x2+5) is

The correct option is -sin (x²+5).2x.

To find the derivative of the function $\cos (x^2+5)$ with respect to $x$, we need to apply the chain rule of differentiation. The chain rule states that for a composite function $f\left(g\right(x\left)\right)$, the derivative is given by:

$$\frac{d}{dx}\left[f\right(g\left(x\right)\left)\right]=f^{\prime }\left(g\right(x\left)\right)\cdot g^{\prime }\left(x\right)$$

Let us break this down into steps:
1. Identify the outer function and the inner function:
The outer function is $f\left(u\right)=\cos \left(u\right)$, where $u=x^2+5$ is the inner function.
2. Differentiate the outer function with respect to its inner argument $u$:
The derivative of $\cos \left(u\right)$ is $-\sin \left(u\right)$. Substituting $u$ back, we get:
$-\sin (x^2+5)$.
3. Differentiate the inner function $g\left(x\right)=x^2+5$ with respect to $x$:
Using the power rule, the derivative of $x^2$ is $2x$ and the derivative of the constant $5$ is $0$. So, the derivative of the inner function is:
$2x$.

Finally, multiply these two results together as prescribed by the chain rule:
$$\frac{d}{dx}\left[\cos \right(x^2+5\left)\right]=-\sin (x^2+5)\cdot 2x$$

Thus, the derivative is $-\sin (x^2+5)\cdot 2x$.