To find the derivative of the function [tex]\( y = x^2 + 3 \)[/tex] with respect to [tex]\( x \)[/tex], we will follow these steps:
1. Identify the function: The function given is [tex]\( y = x^2 + 3 \)[/tex].
2. Basic rules of differentiation:
- The derivative of a constant is 0.
- The power rule states that if [tex]\( y = x^n \)[/tex], then [tex]\( \frac{dy}{dx} = nx^{n-1} \)[/tex].
3. Differentiate each term separately:
- The first term in the function is [tex]\( x^2 \)[/tex].
- Applying the power rule, the derivative of [tex]\( x^2 \)[/tex] is [tex]\( 2x \)[/tex].
- The second term in the function is a constant [tex]\( 3 \)[/tex].
- The derivative of a constant is 0.
4. Combine the results:
- The derivative of [tex]\( y = x^2 + 3 \)[/tex] is the sum of the derivatives of its individual terms.
So, combining these results, we get:
[tex]\[ \frac{dy}{dx} = 2x + 0 \][/tex]
Simplifying this, we have:
[tex]\[ \frac{dy}{dx} = 2x \][/tex]
Therefore, the derivative of [tex]\( y = x^2 + 3 \)[/tex] with respect to [tex]\( x \)[/tex] is [tex]\( 2x \)[/tex].
