Find `(dy)/(dx)` for the functions: `y=x^3e^xsinx`

Explanation

To find \(\frac{dy}{dx}\) for the function \(y = x^3 e^x \sin x\), we will use the product rule of differentiation. The product rule states that if you have a product of three functions \(u\), \(v\), and \(w\), then:

\[ \frac{d}{dx}(uvw) = u'vw + uv'w + uvw' \]

where \(u' = \frac{du}{dx}\), \(v' = \frac{dv}{dx}\), and \(w' = \frac{dw}{dx}\).

### Step 1: Identify the functions
Let:
- \(u = x^3\)
- \(v = e^x\)
- \(w = \sin x\)

### Step 2: Differentiate each function
Now we will differentiate each of these functions:
- \(u' = \frac{d}{dx}(x^3) = 3x^2\)
- \(v' = \frac{d}{dx}(e^x) = e^x\)
- \(w' = \frac{d}{dx}(\sin x) = \cos x\)

### Step 3: Apply the product rule
Now we can apply the product rule:

\[ \frac{dy}{dx} = u'vw + uv'w + uvw' \]

Substituting the values we found:

\[ \frac{dy}{dx} = (3x^2)(e^x)(\sin x) + (x^3)(e^x)(\sin x) + (x^3)(e^x)(\cos x) \]

### Step 4: Factor out common terms
We can factor out the common terms \(x^2 e^x\):

\[ \frac{dy}{dx} = x^2 e^x \left(3 \sin x + x^3 \sin x + x^3 \cos x\right) \]

### Final Answer
Thus, the derivative \(\frac{dy}{dx}\) is:

\[ \frac{dy}{dx} = x^2 e^x \left(3 \sin x + x^3 \sin x + x^3 \cos x\right) \]

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