1. **Stating the problem:** Find the first derivative $\frac{dy}{dx}$ of the function $y = x^3 + 5x^2 - 17$. 2. **Formula and rules:** The derivative of $x^n$ with respect to $x$ is $nx^{n-1}$. Constants differentiate to zero. 3. **Apply the derivative:** $$\frac{dy}{dx} = \frac{d}{dx}(x^3) + \frac{d}{dx}(5x^2) - \frac{d}{dx}(17)$$ 4. **Calculate each term:** $$\frac{d}{dx}(x^3) = 3x^{3-1} = 3x^2$$ $$\frac{d}{dx}(5x^2) = 5 \times 2x^{2-1} = 10x$$ $$\frac{d}{dx}(17) = 0$$ 5. **Combine results:** $$\frac{dy}{dx} = 3x^2 + 10x + 0 = 3x^2 + 10x$$ **Final answer:** $$\boxed{\frac{dy}{dx} = 3x^2 + 10x}$$