To find the value of [tex]\(\frac{dy}{dx}\)[/tex] at [tex]\(x = \frac{\pi}{2}\)[/tex] given the function [tex]\(y = \frac{x^2}{\pi} + 4 \cos x\)[/tex], follow these steps:
1. Define the function:
[tex]\[
y = \frac{x^2}{\pi} + 4 \cos x
\][/tex]
2. Find the derivative of [tex]\(y\)[/tex] with respect to [tex]\(x\)[/tex].
Start by differentiating each term separately.
- Differentiate [tex]\(\frac{x^2}{\pi}\)[/tex]:
[tex]\[
\frac{d}{dx} \left( \frac{x^2}{\pi} \right) = \frac{2x}{\pi}
\][/tex]
- Differentiate [tex]\(4 \cos x\)[/tex]:
[tex]\[
\frac{d}{dx} (4 \cos x) = 4 \cdot (-\sin x) = -4 \sin x
\][/tex]
3. Combine the derivatives:
[tex]\[
\frac{dy}{dx} = \frac{2x}{\pi} - 4 \sin x
\][/tex]
4. Evaluate the derivative at [tex]\(x = \frac{\pi}{2}\)[/tex]:
Substitute [tex]\(x = \frac{\pi}{2}\)[/tex] into the derivative:
[tex]\[
\frac{dy}{dx} \bigg|_{x = \frac{\pi}{2}} = \frac{2 \left( \frac{\pi}{2} \right)}{\pi} - 4 \sin \left( \frac{\pi}{2} \right)
\][/tex]
5. Simplify the expression:
[tex]\[
\frac{dy}{dx} \bigg|_{x = \frac{\pi}{2}} = \frac{2 \cdot \frac{\pi}{2}}{\pi} - 4 \cdot 1
\][/tex]
[tex]\[
= \frac{\pi}{\pi} - 4
\][/tex]
[tex]\[
= 1 - 4
\][/tex]
[tex]\[
= -3
\][/tex]
Therefore, the value of [tex]\(\frac{dy}{dx}\)[/tex] at [tex]\(x = \frac{\pi}{2}\)[/tex] is [tex]\(-3\)[/tex].
