If \[{\text{f}}\left( {\text{x}} \right) = \sin \left| {\text{x}} \right|\] then value of \[\frac{{{\text{df}}}}{{{\text{dx}}}}\] at \[{\text{x}} = \frac{{ – \pi }}{4}\] is A. 0 B. \[\frac{1}{{\sqrt 2 }}\] C. \[ – \frac{1}{{\sqrt 2 }}\] D. 1

The correct answer is $\boxed{0}$.

The derivative of $\sin\left| x \right|$ is $\cos\left| x \right|$. Since $\cos\left( \frac{{ – \pi }}{4} \right) = 0$, the value of $\frac{{{\text{df}}}}{{{\text{dx}}}}$ at $\frac{{ – \pi }}{4}$ is $0$.

Here is a step-by-step solution:

1. Let $f(x) = \sin |x|$.
2. The derivative of $f$ is $f'(x) = \cos |x|$.
3. Substitute $x = \frac{-\pi}{4}$ into $f’$.
4. $\cos\left( \frac{{ – \pi }}{4} \right) = 0$.
5. Therefore, the value of $\frac{{{\text{df}}}}{{{\text{dx}}}}$ at $\frac{{ – \pi }}{4}$ is $0$.