Differentiate the function with respect to $x$: $\sin(x^{2}+5)$.

Let $f(x) = \sin(x^{2}+5)$.
Using the chain rule,we differentiate the outer function $\sin(u)$ and multiply it by the derivative of the inner function $u = x^{2}+5$.
$\frac{d}{dx}[\sin(x^{2}+5)] = \cos(x^{2}+5) \cdot \frac{d}{dx}(x^{2}+5)$.
Since $\frac{d}{dx}(x^{2}) = 2x$ and $\frac{d}{dx}(5) = 0$,we have $\frac{d}{dx}(x^{2}+5) = 2x$.
Therefore,$\frac{d}{dx}[\sin(x^{2}+5)] = \cos(x^{2}+5) \cdot 2x = 2x \cos(x^{2}+5)$.