What is the derivative of x³ + 2x with respect to x? Give a full worked solution with steps.

We can differentiate each term separately:
$\frac{d}{dx}(x^3 + 2x) = \frac{d}{dx}(x^3) + \frac{d}{dx}(2x)$

Using the Power Rule where $n=3$:
$\frac{d}{dx}(x^3) = 3x^{3-1} = 3x^2$

Using the Constant Multiple Rule and the Power Rule (where $x = x^1$, so $n=1$):
$\frac{d}{dx}(2x) = 2 \cdot \frac{d}{dx}(x^1)$
$\frac{d}{dx}(x^1) = 1x^{1-1} = 1x^0 = 1 \cdot 1 = 1$
So, $\frac{d}{dx}(2x) = 2 \cdot 1 = 2$

Substitute the derivatives of each term back into the expression from Step 1:
$\frac{d}{dx}(x^3 + 2x) = 3x^2 + 2$