Answer :
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].
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].
