Find the derivative of x³ sin x

Step-by-step solution

1. 1

   Given Function

   We need to find the derivative of $y = x^3 \sin x$.

   $$y = x^3 \sin x$$
2. 2

   Product Rule

   When differentiating a product of two functions, we use the product rule.

   $$\frac{d}{dx}(uv) = u'v + uv'$$
3. 3

   Identify u and v

   Let $u = x^3$ and $v = \sin x$.

   $$u = x^3, v = \sin x$$
4. 4

   Find u' and v'

   Differentiate $u$ and $v$ with respect to $x$.

   $$u' = \frac{d}{dx}(x^3) = 3x^2, v' = \frac{d}{dx}(\sin x) = \cos x$$
5. 5

   Apply Product Rule

   Substitute $u, v, u', v'$ into the product rule formula.

   $$\frac{dy}{dx} = (3x^2)(\sin x) + (x^3)(\cos x)$$
6. 6

   Final Derivative

   The derivative of $x^3 \sin x$ is $3x^2 \sin x + x^3 \cos x$.

   $$\frac{dy}{dx} = 3x^2 \sin x + x^3 \cos x$$