1. **State the problem:** We are given the curve equation $$y = 2x^3 - x^2 + ax - 5$$ where $$a$$ is a constant. 2. **Find the derivative $$\frac{dy}{dx}$$:** To find the slope of the curve at any point, we differentiate $$y$$ with respect to $$x$$. 3. **Recall differentiation rules:** - The derivative of $$x^n$$ is $$nx^{n-1}$$. - The derivative of a constant times a function is the constant times the derivative of the function. - The derivative of a constant is zero. 4. **Apply the rules term-by-term:** - Derivative of $$2x^3$$ is $$2 \times 3x^{3-1} = 6x^2$$. - Derivative of $$-x^2$$ is $$-2x$$. - Derivative of $$ax$$ is $$a$$ (since $$a$$ is constant and derivative of $$x$$ is 1). - Derivative of $$-5$$ is $$0$$. 5. **Combine all derivatives:** $$\frac{dy}{dx} = 6x^2 - 2x + a$$ **Final answer:** $$\boxed{\frac{dy}{dx} = 6x^2 - 2x + a}$$