The derivative of $f(x) = 5x^3 + 2x^2 - x + 7$ is:

$$f'(x) = 15x^2 + 4x - 1$$

To find the derivative of the function:

$$f(x) = 5x^3 + 2x^2 - x + 7$$

we differentiate each term separately:

$$\frac{d}{dx} (5x^3) = 15x^2$$
$$\frac{d}{dx} (2x^2) = 4x$$
$$\frac{d}{dx} (-x) = -1$$
$$\frac{d}{dx} (7) = 0$$

So, the derivative is:

$$f'(x) = 15x^2 + 4x - 1$$