What is the derivative of f(x) = x^3 + 2x?

Definition

The derivative of a function $f(x)$ measures the rate at which $f(x)$ changes with respect to $x$. For a function $f(x) = x^n$, the power rule states:

$$ \frac{d}{dx} x^n = n x^{n-1} $$

Worked Example

Given:

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

Step 1: Differentiate each term separately

• For $x^3$:

$$ \frac{d}{dx} x^3 = 3x^{2} $$

• For $2x$: %%MATH_DISPLAY_3%% **Step 2: Combine the results** %%MATH_DISPLAY_4%% ## Takeaways - The derivative of $x^n$ is $n x^{n-1}$ (power rule). - Differentiate each term of a \sum separately. - For $f(x) = x^3 + 2x$, the derivative is $f'(x) = 3x^2 + 2$.