1. Let's start by stating a common problem: Find the derivative of the function $f(x) = x^3 + 5x^2 - 4x + 7$. 2. The formula for the derivative of a power function $x^n$ is given by the power rule: $$\frac{d}{dx} x^n = n x^{n-1}$$. 3. We apply the derivative to each term separately using the linearity of differentiation: $$\frac{d}{dx} (x^3) = 3x^2$$, $$\frac{d}{dx} (5x^2) = 5 \cdot 2x = 10x$$, $$\frac{d}{dx} (-4x) = -4$$, and $$\frac{d}{dx} (7) = 0$$ because the derivative of a constant is zero. 4. Putting it all together, the derivative is: $$f'(x) = 3x^2 + 10x - 4$$. 5. This means the slope of the function $f(x)$ at any point $x$ is given by $3x^2 + 10x - 4$. This completes the solution for the derivative of the given function.