What is the derivative of x² + 3x + 2?

The derivative of a function tells us how the function's value changes as the input $x$ changes. For polynomials like $x^2 + 3x + 2$, we use the power rule for differentiation. The power rule says that the derivative of $x^n$ is $nx^{n-1}$. For constants (numbers without $x$), the derivative is zero because they don't change with $x$. So, for $x^2$, the derivative is $2x$. For $3x$, the derivative is $3$. For $2$, the derivative is $0$. Adding these results gives the derivative of the whole function.

Imagine you have a curve that shows how something changes, like how fast a car is going at different times. The derivative helps us find the speed at any exact moment. Here, the curve is made by the formula $x^2 + 3x + 2$. To find the derivative, we look at how this formula changes when $x$ changes a little bit. We do this by using simple rules for each part of the formula.

Given the function:

$f(x) = x^2 + 3x + 2$

Step 1: Differentiate $x^2$ using the power rule:

$\frac{d}{dx} x^2 = 2x$

Step 2: Differentiate $3x$:

$\frac{d}{dx} 3x = 3$

Step 3: Differentiate constant $2$:

$\frac{d}{dx} 2 = 0$

Step 4: Add all derivatives:

$f'(x) = 2x + 3 + 0 = 2x + 3$

Final answer:

$\boxed{2x + 3}$