1. We need to find the derivative of the function $$f(x)=x^5\cdot \sin(x)\cdot \cos(x)$$. 2. This is a product of three functions: $u=x^5$, $v=\sin(x)$, and $w=\cos(x)$. We will use the product rule for three functions: $$\frac{d}{dx}(u\cdot v \cdot w) = u'\cdot v \cdot w + u \cdot v' \cdot w + u \cdot v \cdot w'$$ 3. Compute the derivatives: $$u' = \frac{d}{dx}(x^5) = 5x^4$$ $$v' = \frac{d}{dx}(\sin(x)) = \cos(x)$$ $$w' = \frac{d}{dx}(\cos(x)) = -\sin(x)$$ 4. Substitute these into the product rule formula: $$f'(x) = 5x^4 \cdot \sin(x) \cdot \cos(x) + x^5 \cdot \cos(x) \cdot \cos(x) + x^5 \cdot \sin(x) \cdot (-\sin(x))$$ 5. Simplify the expression: $$f'(x) = 5x^4 \sin(x) \cos(x) + x^5 \cos^2(x) - x^5 \sin^2(x)$$ 6. Recognize the identity $$\cos^2(x) - \sin^2(x) = \cos(2x)$$, so: $$f'(x) = 5x^4 \sin(x) \cos(x) + x^5 \cos(2x)$$ Final answer: $$\boxed{f'(x) = 5x^4 \sin(x) \cos(x) + x^5 \cos(2x)}$$