1. **State the problem:** We are given the function $$y = 7x - \sin^2(x)$$ and need to find its differential $$dy$$. 2. **Recall the formula for differential:** The differential of a function $$y = f(x)$$ is given by $$dy = f'(x) dx$$, where $$f'(x)$$ is the derivative of $$f(x)$$ with respect to $$x$$. 3. **Differentiate each term:** - The derivative of $$7x$$ is $$7$$. - The term $$\sin^2(x)$$ can be seen as $$(\sin(x))^2$$. Using the chain rule: $$\frac{d}{dx} \sin^2(x) = 2 \sin(x) \cdot \cos(x)$$. 4. **Combine derivatives:** $$f'(x) = 7 - 2 \sin(x) \cos(x)$$. 5. **Write the differential:** $$dy = \left(7 - 2 \sin(x) \cos(x)\right) dx$$. 6. **Summary:** The differential $$dy$$ expresses how $$y$$ changes with a small change in $$x$$, calculated using the derivative multiplied by $$dx$$.