Evaluate the derivative when $x$ is equal to a value

Question

For the derivative $f^{\prime}() = \cos - x\sin(x)$. I need to evaluate it when $x=\frac{\pi}{4}$.

The textbook shows the result as : $f^{\prime}(\frac{\pi}{4})=\frac{\sqrt{2}}{8}(4−\pi)$

Unfortunately, I don't know how to get to that result.

Can someone show me step by step? Thank you so much in advance!

Answer 1

When $x=\frac \pi 4$ you have $\cos(x)=\sin(x)=\frac {\sqrt 2}{2}$, so

$\cos(x) - x\sin(x) \\=\frac {\sqrt 2}{2}-\frac \pi 4 \times\frac {\sqrt 2}{2} \\= \frac {\sqrt 2}{2} \left(1-\frac \pi 4\right) \\= \frac {\sqrt 2}{8}(4-\pi)$