Derivative of a polynomial using linearity rules

Differentiate each term separately using the sum/difference rule: $f'(x) = \frac{d}{dx}[5x^3] - \frac{d}{dx}[4x^2] + \frac{d}{dx}[7x] - \frac{d}{dx}[2]$. For each term, apply the power rule $\frac{d}{dx}[x^n] = nx^{n-1}$ together with the constant multiple rule $\frac{d}{dx}[cf(x)] = cf'(x)$.

Computing each piece: $\frac{d}{dx}[5x^3] = 5(3x^2) = 15x^2$, $\frac{d}{dx}[4x^2] = 4(2x) = 8x$, $\frac{d}{dx}[7x] = 7$, and $\frac{d}{dx}[2] = 0$ since the derivative of a constant is zero. Combining these gives $f'(x) = 15x^2 - 8x + 7$.