Solve $\frac{d}{dx}\left(x^3\sin x\right)$

Step-by-step solution

1. Step 1 · Identify the structure
   $$f=x^3,g=\sin x$$

   The expression is a product, so the product rule applies.

2. Step 2 · Differentiate each factor
   $$f^{\prime }=3x^2,g^{\prime }=\cos x$$

   Power rule for x³ and the standard derivative of sine.

3. Step 3 · Product rule
   $$3x^2\sin x+x^3\cos x$$

   (fg)' = f'g + fg'.