product rule\nrelated content\n$\frac{d}{dx}(x^{3}sin(x))=$

Explanation:

Step1: Recall product - rule

The product - rule states that if (y = u\cdot v), then (y^\prime=\frac{d}{dx}(u\cdot v)=u^\prime v + uv^\prime). Let (u = x^{3}) and (v=\sin(x)).

Step2: Differentiate (u) and (v)

Differentiate (u = x^{3}) with respect to (x): (\frac{d}{dx}(x^{3})=3x^{2}) (using the power - rule (\frac{d}{dx}(x^{n})=nx^{n - 1})). Differentiate (v=\sin(x)) with respect to (x): (\frac{d}{dx}(\sin(x))=\cos(x)).

Step3: Apply product - rule

Substitute (u), (u^\prime), (v), and (v^\prime) into the product - rule formula. (\frac{d}{dx}(x^{3}\sin(x))=\frac{d}{dx}(x^{3})\sin(x)+x^{3}\frac{d}{dx}(\sin(x))=3x^{2}\sin(x)+x^{3}\cos(x)).

Answer:

(3x^{2}\sin(x)+x^{3}\cos(x))