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Answer

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Step 1:
I will solve the following calculus problem:

Find the derivative of the function $$f(x) = x^3 \sin(x)$$.

Step 2:
: Recall the product rule for differentiation.

In this case, let $$g(x) = x^3$$ and $$h(x) = \sin(x)$$.

Step 3:
: Find the derivatives of g(x) and h(x).

h'(x) = \cos(x)
We know that: and

Step 4:
: Plug these expressions into the product rule formula from Step 1:

f'(x) = (3x^2)(\sin(x)) + (x^3)(\cos(x))

Step 5:
: Simplify the expression:

f'(x) = 3x^2 \sin(x) + x^3 \cos(x)

Final Answer

The derivative of the function f(x) = x^3 \sin(x) is f'(x) = 3x^2 \sin(x) + x^3 \cos(x).