Find the derivative of $x^2 \sin x$.

How we read your problem

Find the derivative of x²·sin(x) with respect to x.

Step-by-step solution

1. 1

   Product rule: (uv)′ = u′v + uv′

   $$\frac{d}{dx}\left[{x}^{2} \sin\left( x\right)\right] = 2x \sin\left( x\right)+{x}^{2} \cos\left( x\right)$$
2. 2

   d/dx[sin x] = cos x

   $$\frac{d}{dx}\left[\sin\left( x\right)\right] = \cos\left( x\right)$$
3. 3

   Power rule: d/dx[xⁿ] = n·xⁿ⁻¹

   $$\frac{d}{dx}\left[{x}^{2}\right] = 2x$$
4. 4

   Combine and simplify

   $$f'(x) = 2x \sin\left( x\right)+{x}^{2} \cos\left( x\right)$$

Formulas used

• Power rule

  $$\frac{d}{dx}x^n = n x^{n-1}$$

• Product rule

  $$(uv)' = u'v + uv'$$

The derivative is built from the outside in: sums split into terms, constants factor out, products and quotients use their rules, and every composite function multiplies by the derivative of its inside (chain rule).

Verification

• Cross-checked numerically against an independent symbolic differentiator at 8 sample points ✓