1. The problem states: If $$\frac{d}{dx} \sin(5x + b) = a \cos(5x + 4)$$, find the value of $$(a + b)$$. 2. Recall the derivative formula for sine: $$\frac{d}{dx} \sin(u) = \cos(u) \cdot \frac{du}{dx}$$. 3. Here, $$u = 5x + b$$, so $$\frac{du}{dx} = 5$$. 4. Therefore, $$\frac{d}{dx} \sin(5x + b) = 5 \cos(5x + b)$$. 5. Given that $$\frac{d}{dx} \sin(5x + b) = a \cos(5x + 4)$$, we equate: $$5 \cos(5x + b) = a \cos(5x + 4)$$. 6. For this equality to hold for all $$x$$, the arguments of cosine must be equal or differ by multiples of $$2\pi$$, and the coefficients must be equal: $$b = 4$$ $$a = 5$$ 7. Finally, compute $$a + b = 5 + 4 = 9$$. Answer: 9